Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

·>Uµ W =12

∫ϕρf dτ

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

·>Uµ W =12

∫ϕρf dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

·>Uµ W =12

∫ϕρf dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=12

∑i

∮Si

ϕiσqidσi

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

·>Uµ W =12

∫ϕρf dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=12

∑i

∮Si

ϕiσqidσi σqi ��N i �¡>Ö�Ý

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

·>Uµ W =12

∫ϕρf dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=12

∑i

∮Si

ϕiσqidσi σqi ��N i �¡>Ö�Ý

=12

∑i

ϕi

∮Si

σqidσi

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

·>Uµ W =12

∫ϕρf dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=12

∑i

∮Si

ϕiσqidσi σqi ��N i �¡>Ö�Ý

=12

∑i

ϕi

∮Si

σqidσi |^�N´�³N§ϕi �~ê

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

·>Uµ W =12

∫ϕρf dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=12

∑i

∮Si

ϕiσqidσi σqi ��N i �¡>Ö�Ý

=12

∑i

ϕi

∮Si

σqidσi |^�N´�³N§ϕi �~ê

=12

∑i

ϕiQi

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

·>Uµ W =12

∫ϕρf dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=12

∑i

∮Si

ϕiσqidσi σqi ��N i �¡>Ö�Ý

=12

∑i

ϕi

∮Si

σqidσi |^�N´�³N§ϕi �~ê

=12

∑i

ϕiQi Qi ��N i �>þ

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

§ 3.3 ���NNNXXX

�!�NX�·>U

·>Uµ W =12

∫ϕρf dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=12

∑i

∮Si

ϕiσqidσi σqi ��N i �¡>Ö�Ý

=12

∑i

ϕi

∮Si

σqidσi |^�N´�³N§ϕi �~ê

=12

∑i

ϕiQi Qi ��N i �>þ

�NX�·>UµW =12

∑i

ϕiQi

E��Æ ÔnX ��� Mï� 1

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

¡>Ö©Ù σ11 Ú σ21 3�N 1 Ú�N 2 þ�>³�~ê p11 Ú p21

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

¡>Ö©Ù σ11 Ú σ21 3�N 1 Ú�N 2 þ�>³�~ê p11 Ú p21

e�N 1 �> Q1§@où��N 1!�N 2 �¡>Ö©Ù�õ�º

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

¡>Ö©Ù σ11 Ú σ21 3�N 1 Ú�N 2 þ�>³�~ê p11 Ú p21

e�N 1 �> Q1§@où��N 1!�N 2 �¡>Ö©Ù�õ�º Q1σ11!Q1σ21

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

¡>Ö©Ù σ11 Ú σ21 3�N 1 Ú�N 2 þ�>³�~ê p11 Ú p21

e�N 1 �> Q1§@où��N 1!�N 2 �¡>Ö©Ù�õ�º Q1σ11!Q1σ21

Ï�ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 Ú Q1p21§

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

¡>Ö©Ù σ11 Ú σ21 3�N 1 Ú�N 2 þ�>³�~ê p11 Ú p21

e�N 1 �> Q1§@où��N 1!�N 2 �¡>Ö©Ù�õ�º Q1σ11!Q1σ21

Ï�ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 Ú Q1p21§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú 0"

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

¡>Ö©Ù σ11 Ú σ21 3�N 1 Ú�N 2 þ�>³�~ê p11 Ú p21

e�N 1 �> Q1§@où��N 1!�N 2 �¡>Ö©Ù�õ�º Q1σ11!Q1σ21

Ï�ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 Ú Q1p21§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú 0"

aq/µe�N 2 �ü >þ§�N 1 Ø�>"�N 1!�N 2 �¡>Ö©Ù©O� σ12!σ22

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

¡>Ö©Ù σ11 Ú σ21 3�N 1 Ú�N 2 þ�>³�~ê p11 Ú p21

e�N 1 �> Q1§@où��N 1!�N 2 �¡>Ö©Ù�õ�º Q1σ11!Q1σ21

Ï�ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 Ú Q1p21§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú 0"

aq/µe�N 2 �ü >þ§�N 1 Ø�>"�N 1!�N 2 �¡>Ö©Ù©O� σ12!σ22

¡>Ö©Ù σ12 Ú σ22 3�N 1 Ú�N 2 þ�>³�~ê p12 Ú p22

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

¡>Ö©Ù σ11 Ú σ21 3�N 1 Ú�N 2 þ�>³�~ê p11 Ú p21

e�N 1 �> Q1§@où��N 1!�N 2 �¡>Ö©Ù�õ�º Q1σ11!Q1σ21

Ï�ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 Ú Q1p21§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú 0"

aq/µe�N 2 �ü >þ§�N 1 Ø�>"�N 1!�N 2 �¡>Ö©Ù©O� σ12!σ22

¡>Ö©Ù σ12 Ú σ22 3�N 1 Ú�N 2 þ�>³�~ê p12 Ú p22

e�N 2 �> Q2§@où��N 1!�N 2 �¡>Ö©Ù�µQ2σ12!Q2σ22

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�!�NX�>³Xê

�50�¥��NX§e1 j ��Nþ�o>þ� Qj§j = 1, 2, . . . , n £n��Nê8¤§K1 i ��Nþ�>³�L�µ

ϕi =∑

j

pijQj

pij ¡��NX�>³Xê (coefficients of potential)§���N�/GÚ 

�k'§��Nþ�>þ k'ºÃ'º Ã'�

Ônþ�n)µ b�kü��N

�N 1 �ü >þ§�N 2 Ø�>"ù��N 1!�N 2 �¡>Ö©Ù©O� σ11!σ21

¡>Ö©Ù σ11 Ú σ21 3�N 1 Ú�N 2 þ�>³�~ê p11 Ú p21

e�N 1 �> Q1§@où��N 1!�N 2 �¡>Ö©Ù�õ�º Q1σ11!Q1σ21

Ï�ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 Ú Q1p21§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú 0"

aq/µe�N 2 �ü >þ§�N 1 Ø�>"�N 1!�N 2 �¡>Ö©Ù©O� σ12!σ22

¡>Ö©Ù σ12 Ú σ22 3�N 1 Ú�N 2 þ�>³�~ê p12 Ú p22

e�N 2 �> Q2§@où��N 1!�N 2 �¡>Ö©Ù�µQ2σ12!Q2σ22

y4�N 1 �> Q1 �Ó� �N 2 �> Q2§XÛº

E��Æ ÔnX ��� Mï� 2

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

�N 1 þ�>³µϕ1 = p11Q1 + p12Q2¶�N 2 þ�>³µϕ2 = p21Q1 + p22Q2"

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

�N 1 þ�>³µϕ1 = p11Q1 + p12Q2¶�N 2 þ�>³µϕ2 = p21Q1 + p22Q2"

>³Xêµp11, p12, p21, p22 ��Nþ�>þ Q1!Q2 Ã'"

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

�N 1 þ�>³µϕ1 = p11Q1 + p12Q2¶�N 2 þ�>³µϕ2 = p21Q1 + p22Q2"

>³Xêµp11, p12, p21, p22 ��Nþ�>þ Q1!Q2 Ã'"

pij �¿Âµ��NX¥=k1 j ��N�ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

�N 1 þ�>³µϕ1 = p11Q1 + p12Q2¶�N 2 þ�>³µϕ2 = p21Q1 + p22Q2"

>³Xêµp11, p12, p21, p22 ��Nþ�>þ Q1!Q2 Ã'"

pij �¿Âµ��NX¥=k1 j ��N�ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

��y²µ

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

�N 1 þ�>³µϕ1 = p11Q1 + p12Q2¶�N 2 þ�>³µϕ2 = p21Q1 + p22Q2"

>³Xêµp11, p12, p21, p22 ��Nþ�>þ Q1!Q2 Ã'"

pij �¿Âµ��NX¥=k1 j ��N�ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

��y²µ

b��NX�k1 i ��N�> 1§ù��m�:�>³� pi(~r)§

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

�N 1 þ�>³µϕ1 = p11Q1 + p12Q2¶�N 2 þ�>³µϕ2 = p21Q1 + p22Q2"

>³Xêµp11, p12, p21, p22 ��Nþ�>þ Q1!Q2 Ã'"

pij �¿Âµ��NX¥=k1 j ��N�ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

��y²µ

b��NX�k1 i ��N�> 1§ù��m�:�>³� pi(~r)§pi(~r) ÷v½)^�µ

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

�N 1 þ�>³µϕ1 = p11Q1 + p12Q2¶�N 2 þ�>³µϕ2 = p21Q1 + p22Q2"

>³Xêµp11, p12, p21, p22 ��Nþ�>þ Q1!Q2 Ã'"

pij �¿Âµ��NX¥=k1 j ��N�ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

��y²µ

b��NX�k1 i ��N�> 1§ù��m�:�>³� pi(~r)§pi(~r) ÷v½)^�µ

(1) 31 j ��Nþ>³�~ê pji§j = 1, 2, 3, . . .

(2) ε

∮Sj

∂pi(~r)

∂ndσ =

{−1 j = i

0 j 6= i

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

�N 1 þ�>³µϕ1 = p11Q1 + p12Q2¶�N 2 þ�>³µϕ2 = p21Q1 + p22Q2"

>³Xêµp11, p12, p21, p22 ��Nþ�>þ Q1!Q2 Ã'"

pij �¿Âµ��NX¥=k1 j ��N�ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

��y²µ

b��NX�k1 i ��N�> 1§ù��m�:�>³� pi(~r)§pi(~r) ÷v½)^�µ

(1) 31 j ��Nþ>³�~ê pji§j = 1, 2, 3, . . .

(2) ε

∮Sj

∂pi(~r)

∂ndσ =

{−1 j = i

0 j 6= i

y41 i ��N�> Qi§ù��m�:>³AT� Qipi(~r)§Ï�§÷v½)^�µ

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

ù��N 1 >Ö©ÙµQ1σ11 + Q2σ12§�N 2 >Ö©ÙµQ1σ21 + Q2σ22§

ù«>Ö©Ù�yµ1. �N 1 Ú�N 2 þ�>³�~ê Q1p11 + Q2p12 Ú Q1p21 + Q2p22§

2. �N 1 Ú�N 2 þ�>þ©O�µQ1 Ú Q2"

��5½n�yù´���(�)"Ïd§�N 1 �> Q1 �Ó� �N 2 �> Q2 �§

�N 1 þ�>³µϕ1 = p11Q1 + p12Q2¶�N 2 þ�>³µϕ2 = p21Q1 + p22Q2"

>³Xêµp11, p12, p21, p22 ��Nþ�>þ Q1!Q2 Ã'"

pij �¿Âµ��NX¥=k1 j ��N�ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

��y²µ

b��NX�k1 i ��N�> 1§ù��m�:�>³� pi(~r)§pi(~r) ÷v½)^�µ

(1) 31 j ��Nþ>³�~ê pji§j = 1, 2, 3, . . .

(2) ε

∮Sj

∂pi(~r)

∂ndσ =

{−1 j = i

0 j 6= i

y41 i ��N�> Qi§ù��m�:>³AT� Qipi(~r)§Ï�§÷v½)^�µ

(1) 31 j ��Nþ>³�~ê pjiQi§j = 1, 2, 3, . . .

(2) ε

∮Sj

∂[Qipi(~r)]

∂ndσ =

{−Qi j = i

0 j 6= i

E��Æ ÔnX ��� Mï� 3

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

ù� Qipi(~r) + Qkpk(~r) ÷v½)^�µ £U\�n¤

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

ù� Qipi(~r) + Qkpk(~r) ÷v½)^�µ £U\�n¤

(1) 31 j ��Nþ>³�~ê pjiQi + pjkQk§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[Qipi(~r) + Qkpk(~r)]

∂ndσ =

−Qi j = i

−Qk j = k

0 j 6= i, k

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

ù� Qipi(~r) + Qkpk(~r) ÷v½)^�µ £U\�n¤

(1) 31 j ��Nþ>³�~ê pjiQi + pjkQk§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[Qipi(~r) + Qkpk(~r)]

∂ndσ =

−Qi j = i

−Qk j = k

0 j 6= i, k

�1 i ��N� Qi �1 k ��N� Qk£Ù{Ø�>¤�§�m>³�µ

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

ù� Qipi(~r) + Qkpk(~r) ÷v½)^�µ £U\�n¤

(1) 31 j ��Nþ>³�~ê pjiQi + pjkQk§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[Qipi(~r) + Qkpk(~r)]

∂ndσ =

−Qi j = i

−Qk j = k

0 j 6= i, k

�1 i ��N� Qi �1 k ��N� Qk£Ù{Ø�>¤�§�m>³�µQipi(~r) + Qkpk(~r)

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

ù� Qipi(~r) + Qkpk(~r) ÷v½)^�µ £U\�n¤

(1) 31 j ��Nþ>³�~ê pjiQi + pjkQk§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[Qipi(~r) + Qkpk(~r)]

∂ndσ =

−Qi j = i

−Qk j = k

0 j 6= i, k

�1 i ��N� Qi �1 k ��N� Qk£Ù{Ø�>¤�§�m>³�µQipi(~r) + Qkpk(~r)

y3§41 l ��N�> Ql, l = 1, 2, . . . , n§ù�∑

l

Qlpl(~r) ÷v½)^�µ

(1) 31 j ��Nþ>³�~ê∑

l pjlQl§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[∑

l Qlpl(~r)]

∂ndσ = −Qj

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

ù� Qipi(~r) + Qkpk(~r) ÷v½)^�µ £U\�n¤

(1) 31 j ��Nþ>³�~ê pjiQi + pjkQk§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[Qipi(~r) + Qkpk(~r)]

∂ndσ =

−Qi j = i

−Qk j = k

0 j 6= i, k

�1 i ��N� Qi �1 k ��N� Qk£Ù{Ø�>¤�§�m>³�µQipi(~r) + Qkpk(~r)

y3§41 l ��N�> Ql, l = 1, 2, . . . , n§ù�∑

l

Qlpl(~r) ÷v½)^�µ

(1) 31 j ��Nþ>³�~ê∑

l pjlQl§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[∑

l Qlpl(~r)]

∂ndσ = −Qj

Ïd1 l ��N�> Ql � (l = 1, 2, . . . , n)§�m>³A�µ

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

ù� Qipi(~r) + Qkpk(~r) ÷v½)^�µ £U\�n¤

(1) 31 j ��Nþ>³�~ê pjiQi + pjkQk§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[Qipi(~r) + Qkpk(~r)]

∂ndσ =

−Qi j = i

−Qk j = k

0 j 6= i, k

�1 i ��N� Qi �1 k ��N� Qk£Ù{Ø�>¤�§�m>³�µQipi(~r) + Qkpk(~r)

y3§41 l ��N�> Ql, l = 1, 2, . . . , n§ù�∑

l

Qlpl(~r) ÷v½)^�µ

(1) 31 j ��Nþ>³�~ê∑

l pjlQl§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[∑

l Qlpl(~r)]

∂ndσ = −Qj

Ïd1 l ��N�> Ql � (l = 1, 2, . . . , n)§�m>³A�µ∑

l

Qlpl(~r)

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

ù� Qipi(~r) + Qkpk(~r) ÷v½)^�µ £U\�n¤

(1) 31 j ��Nþ>³�~ê pjiQi + pjkQk§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[Qipi(~r) + Qkpk(~r)]

∂ndσ =

−Qi j = i

−Qk j = k

0 j 6= i, k

�1 i ��N� Qi �1 k ��N� Qk£Ù{Ø�>¤�§�m>³�µQipi(~r) + Qkpk(~r)

y3§41 l ��N�> Ql, l = 1, 2, . . . , n§ù�∑

l

Qlpl(~r) ÷v½)^�µ

(1) 31 j ��Nþ>³�~ê∑

l pjlQl§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[∑

l Qlpl(~r)]

∂ndσ = −Qj

Ïd1 l ��N�> Ql � (l = 1, 2, . . . , n)§�m>³A�µ∑

l

Qlpl(~r)

�=µ��NX¥��N©O�> Ql, l = 1, 2, . . . , n �§1 i ��Nþ�>³�µ∑

l

pilQl

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

yÓ�41 i ��N�> Qi �1 k ��N�> Qk§Ù§�NØ�>

ù� Qipi(~r) + Qkpk(~r) ÷v½)^�µ £U\�n¤

(1) 31 j ��Nþ>³�~ê pjiQi + pjkQk§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[Qipi(~r) + Qkpk(~r)]

∂ndσ =

−Qi j = i

−Qk j = k

0 j 6= i, k

�1 i ��N� Qi �1 k ��N� Qk£Ù{Ø�>¤�§�m>³�µQipi(~r) + Qkpk(~r)

y3§41 l ��N�> Ql, l = 1, 2, . . . , n§ù�∑

l

Qlpl(~r) ÷v½)^�µ

(1) 31 j ��Nþ>³�~ê∑

l pjlQl§j = 1, 2, . . . , n

(2) ε

∮Sj

∂[∑

l Qlpl(~r)]

∂ndσ = −Qj

Ïd1 l ��N�> Ql � (l = 1, 2, . . . , n)§�m>³A�µ∑

l

Qlpl(~r)

�=µ��NX¥��N©O�> Ql, l = 1, 2, . . . , n �§1 i ��Nþ�>³�µ∑

l

pilQl

pij �¿Âµ��NX¥=k1 j ��N�ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

E��Æ ÔnX ��� Mï� 4

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

XJ�k Q1 UCµ dW =(∂W

∂Q1

)dQ1 = 1

2

∑j

(p1j + pj1)QjdQ1

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

XJ�k Q1 UCµ dW =(∂W

∂Q1

)dQ1 = 1

2

∑j

(p1j + pj1)QjdQ1

,��¡§du�N 1 >þO\ dQ1 ��NX·>U�Cz§

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

XJ�k Q1 UCµ dW =(∂W

∂Q1

)dQ1 = 1

2

∑j

(p1j + pj1)QjdQ1

,��¡§du�N 1 >þO\ dQ1 ��NX·>U�Cz§

�ur dQ1 �>þlá�£��N 1 L§§å�Ñ·>å��õ

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

XJ�k Q1 UCµ dW =(∂W

∂Q1

)dQ1 = 1

2

∑j

(p1j + pj1)QjdQ1

,��¡§du�N 1 >þO\ dQ1 ��NX·>U�Cz§

�ur dQ1 �>þlá�£��N 1 L§§å�Ñ·>å��õ

��uù dQ1 �>Ö·>³U�O\µdU = dQ1(ϕ1 − ϕá�) = dQ1ϕ1

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

XJ�k Q1 UCµ dW =(∂W

∂Q1

)dQ1 = 1

2

∑j

(p1j + pj1)QjdQ1

,��¡§du�N 1 >þO\ dQ1 ��NX·>U�Cz§

�ur dQ1 �>þlá�£��N 1 L§§å�Ñ·>å��õ

��uù dQ1 �>Ö·>³U�O\µdU = dQ1(ϕ1 − ϕá�) = dQ1ϕ1

Ïd dU = ϕ1dQ1 =[∑

j

p1jQj

]dQ1 =

∑j

p1jQjdQ1

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

XJ�k Q1 UCµ dW =(∂W

∂Q1

)dQ1 = 1

2

∑j

(p1j + pj1)QjdQ1

,��¡§du�N 1 >þO\ dQ1 ��NX·>U�Cz§

�ur dQ1 �>þlá�£��N 1 L§§å�Ñ·>å��õ

��uù dQ1 �>Ö·>³U�O\µdU = dQ1(ϕ1 − ϕá�) = dQ1ϕ1

Ïd dU = ϕ1dQ1 =[∑

j

p1jQj

]dQ1 =

∑j

p1jQjdQ1

'� dW Ú dUµ dU =∑

j

p1jQjdQ1 = 12

∑j

(p1j + pj1)QjdQ1 = dW

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

XJ�k Q1 UCµ dW =(∂W

∂Q1

)dQ1 = 1

2

∑j

(p1j + pj1)QjdQ1

,��¡§du�N 1 >þO\ dQ1 ��NX·>U�Cz§

�ur dQ1 �>þlá�£��N 1 L§§å�Ñ·>å��õ

��uù dQ1 �>Ö·>³U�O\µdU = dQ1(ϕ1 − ϕá�) = dQ1ϕ1

Ïd dU = ϕ1dQ1 =[∑

j

p1jQj

]dQ1 =

∑j

p1jQjdQ1

'� dW Ú dUµ dU =∑

j

p1jQjdQ1 = 12

∑j

(p1j + pj1)QjdQ1 = dW

þªé?¿ Qj ¤á p1jQj =1

2(p1j + pj1)Qj

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

XJ�k Q1 UCµ dW =(∂W

∂Q1

)dQ1 = 1

2

∑j

(p1j + pj1)QjdQ1

,��¡§du�N 1 >þO\ dQ1 ��NX·>U�Cz§

�ur dQ1 �>þlá�£��N 1 L§§å�Ñ·>å��õ

��uù dQ1 �>Ö·>³U�O\µdU = dQ1(ϕ1 − ϕá�) = dQ1ϕ1

Ïd dU = ϕ1dQ1 =[∑

j

p1jQj

]dQ1 =

∑j

p1jQjdQ1

'� dW Ú dUµ dU =∑

j

p1jQjdQ1 = 12

∑j

(p1j + pj1)QjdQ1 = dW

þªé?¿ Qj ¤á p1jQj =1

2(p1j + pj1)Qj =⇒ p1j = pj1

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

�±y²>³Xê÷vµ(1) pij = pji, (2) pij > 0, (3) pii > pij

y²µ

(1) �N·>U W =1

2

∑i

ϕiQi =1

2

∑i

[∑j

pijQj

]Qi =

1

2

∑ij

pijQiQj

�N>þUC� dW =(∂W

∂Q1

)dQ1 + · · ·+

(∂W

∂Qn

)dQn

XJ�k Q1 UCµ dW =(∂W

∂Q1

)dQ1 = 1

2

∑j

(p1j + pj1)QjdQ1

,��¡§du�N 1 >þO\ dQ1 ��NX·>U�Cz§

�ur dQ1 �>þlá�£��N 1 L§§å�Ñ·>å��õ

��uù dQ1 �>Ö·>³U�O\µdU = dQ1(ϕ1 − ϕá�) = dQ1ϕ1

Ïd dU = ϕ1dQ1 =[∑

j

p1jQj

]dQ1 =

∑j

p1jQjdQ1

'� dW Ú dUµ dU =∑

j

p1jQjdQ1 = 12

∑j

(p1j + pj1)QjdQ1 = dW

þªé?¿ Qj ¤á p1jQj =1

2(p1j + pj1)Qj =⇒ p1j = pj1

aq�y²µ pij = pji

E��Æ ÔnX ��� Mï� 5

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

d ϕi =∑

j

pijQj§��µ

Qi =∑

j

Cijϕj,Cii ¡�>NXê (coefficients of capacitance)

Cij (i 6= j) ¡�aAXê(coefficients of induction)

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

d ϕi =∑

j

pijQj§��µ

Qi =∑

j

Cijϕj,Cii ¡�>NXê (coefficients of capacitance)

Cij (i 6= j) ¡�aAXê(coefficients of induction)

�±y²>N>aXê÷vµ(1) Cij = Cji, (2) Cii > 0, Cij

∣∣∣i 6=j

< 0

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

d ϕi =∑

j

pijQj§��µ

Qi =∑

j

Cijϕj,Cii ¡�>NXê (coefficients of capacitance)

Cij (i 6= j) ¡�aAXê(coefficients of induction)

�±y²>N>aXê÷vµ(1) Cij = Cji, (2) Cii > 0, Cij

∣∣∣i 6=j

< 0

~ 1µ3¥5�N¥��:>Ö q§å�N¥¥% r§¦�N¥�>³

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

d ϕi =∑

j

pijQj§��µ

Qi =∑

j

Cijϕj,Cii ¡�>NXê (coefficients of capacitance)

Cij (i 6= j) ¡�aAXê(coefficients of induction)

�±y²>N>aXê÷vµ(1) Cij = Cji, (2) Cii > 0, Cij

∣∣∣i 6=j

< 0

~ 1µ3¥5�N¥��:>Ö q§å�N¥¥% r§¦�N¥�>³

�N¥À��N 1§:>ÖÀ��N 2£�»ªu 0 ��N¥¤

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

d ϕi =∑

j

pijQj§��µ

Qi =∑

j

Cijϕj,Cii ¡�>NXê (coefficients of capacitance)

Cij (i 6= j) ¡�aAXê(coefficients of induction)

�±y²>N>aXê÷vµ(1) Cij = Cji, (2) Cii > 0, Cij

∣∣∣i 6=j

< 0

~ 1µ3¥5�N¥��:>Ö q§å�N¥¥% r§¦�N¥�>³

�N¥À��N 1§:>ÖÀ��N 2£�»ªu 0 ��N¥¤

3�N¥£�N 1¤��ü �>Ö§3:>Ö?�>³�µ ϕq =1

4πε0

1

r= p21

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

d ϕi =∑

j

pijQj§��µ

Qi =∑

j

Cijϕj,Cii ¡�>NXê (coefficients of capacitance)

Cij (i 6= j) ¡�aAXê(coefficients of induction)

�±y²>N>aXê÷vµ(1) Cij = Cji, (2) Cii > 0, Cij

∣∣∣i 6=j

< 0

~ 1µ3¥5�N¥��:>Ö q§å�N¥¥% r§¦�N¥�>³

�N¥À��N 1§:>ÖÀ��N 2£�»ªu 0 ��N¥¤

3�N¥£�N 1¤��ü �>Ö§3:>Ö?�>³�µ ϕq =1

4πε0

1

r= p21

pij �é¡5µ p21 = p12

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

d ϕi =∑

j

pijQj§��µ

Qi =∑

j

Cijϕj,Cii ¡�>NXê (coefficients of capacitance)

Cij (i 6= j) ¡�aAXê(coefficients of induction)

�±y²>N>aXê÷vµ(1) Cij = Cji, (2) Cii > 0, Cij

∣∣∣i 6=j

< 0

~ 1µ3¥5�N¥��:>Ö q§å�N¥¥% r§¦�N¥�>³

�N¥À��N 1§:>ÖÀ��N 2£�»ªu 0 ��N¥¤

3�N¥£�N 1¤��ü �>Ö§3:>Ö?�>³�µ ϕq =1

4πε0

1

r= p21

pij �é¡5µ p21 = p12

p12 �¿Â�µ 3:>Ö£�N 2¤?��ü �>Ö§�N¥£�N 1¤þ�>³

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

d ϕi =∑

j

pijQj§��µ

Qi =∑

j

Cijϕj,Cii ¡�>NXê (coefficients of capacitance)

Cij (i 6= j) ¡�aAXê(coefficients of induction)

�±y²>N>aXê÷vµ(1) Cij = Cji, (2) Cii > 0, Cij

∣∣∣i 6=j

< 0

~ 1µ3¥5�N¥��:>Ö q§å�N¥¥% r§¦�N¥�>³

�N¥À��N 1§:>ÖÀ��N 2£�»ªu 0 ��N¥¤

3�N¥£�N 1¤��ü �>Ö§3:>Ö?�>³�µ ϕq =1

4πε0

1

r= p21

pij �é¡5µ p21 = p12

p12 �¿Â�µ 3:>Ö£�N 2¤?��ü �>Ö§�N¥£�N 1¤þ�>³

3:>Ö£�N 2¤?��>Ö q§3�N¥?�>³�µ ϕ�N¥ = p12q =1

4πε0

q

r

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

(2) l pij�¿Âµ 31 j ��Nþ��ü �>Ö£Ù{Ø�>¤�§1 i ��N�>³"

�y² pij > 0 (b½Ã¡�>³� 0)

(3) aq�±y²µ pii > pij (�y²)

d ϕi =∑

j

pijQj§��µ

Qi =∑

j

Cijϕj,Cii ¡�>NXê (coefficients of capacitance)

Cij (i 6= j) ¡�aAXê(coefficients of induction)

�±y²>N>aXê÷vµ(1) Cij = Cji, (2) Cii > 0, Cij

∣∣∣i 6=j

< 0

~ 1µ3¥5�N¥��:>Ö q§å�N¥¥% r§¦�N¥�>³

�N¥À��N 1§:>ÖÀ��N 2£�»ªu 0 ��N¥¤

3�N¥£�N 1¤��ü �>Ö§3:>Ö?�>³�µ ϕq =1

4πε0

1

r= p21

pij �é¡5µ p21 = p12

p12 �¿Â�µ 3:>Ö£�N 2¤?��ü �>Ö§�N¥£�N 1¤þ�>³

3:>Ö£�N 2¤?��>Ö q§3�N¥?�>³�µ ϕ�N¥ = p12q =1

4πε0

q

r�,§^º�{�N´� ϕ�N¥

E��Æ ÔnX ��� Mï� 6

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½n

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

o�±ò Q1 Ú Q2 ©Ù¤ σ1 Ú σ2§¦�0�L¡��³¡§ù��m>|�µ ~E, ~D

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

o�±ò Q1 Ú Q2 ©Ù¤ σ1 Ú σ2§¦�0�L¡��³¡§ù��m>|�µ ~E, ~D

Thomson ½n �ѵ| ~E, ~D �o>U�u| ~E ′ ~D ′ �o>U

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

o�±ò Q1 Ú Q2 ©Ù¤ σ1 Ú σ2§¦�0�L¡��³¡§ù��m>|�µ ~E, ~D

Thomson ½n �ѵ| ~E, ~D �o>U�u| ~E ′ ~D ′ �o>U

y²:

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

o�±ò Q1 Ú Q2 ©Ù¤ σ1 Ú σ2§¦�0�L¡��³¡§ù��m>|�µ ~E, ~D

Thomson ½n �ѵ| ~E, ~D �o>U�u| ~E ′ ~D ′ �o>U

y²: w,µ∇ · ~D ′ = ∇ · ~D = ρ =

{ρ1 0� 1

ρ2 0� 2, ~E = −∇ϕ, ~E ′ = −∇ϕ′,

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

o�±ò Q1 Ú Q2 ©Ù¤ σ1 Ú σ2§¦�0�L¡��³¡§ù��m>|�µ ~E, ~D

Thomson ½n �ѵ| ~E, ~D �o>U�u| ~E ′ ~D ′ �o>U

y²: w,µ∇ · ~D ′ = ∇ · ~D = ρ =

{ρ1 0� 1

ρ2 0� 2, ~E = −∇ϕ, ~E ′ = −∇ϕ′,∮

S1

σ1 ds =

∮S1

σ′1 ds = Q1,

∮S2

σ2 ds =

∮S2

σ′2 ds = Q2

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

o�±ò Q1 Ú Q2 ©Ù¤ σ1 Ú σ2§¦�0�L¡��³¡§ù��m>|�µ ~E, ~D

Thomson ½n �ѵ| ~E, ~D �o>U�u| ~E ′ ~D ′ �o>U

y²: w,µ∇ · ~D ′ = ∇ · ~D = ρ =

{ρ1 0� 1

ρ2 0� 2, ~E = −∇ϕ, ~E ′ = −∇ϕ′,∮

S1

σ1 ds =

∮S1

σ′1 ds = Q1,

∮S2

σ2 ds =

∮S2

σ′2 ds = Q2 d?± ds L«¡È©

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

o�±ò Q1 Ú Q2 ©Ù¤ σ1 Ú σ2§¦�0�L¡��³¡§ù��m>|�µ ~E, ~D

Thomson ½n �ѵ| ~E, ~D �o>U�u| ~E ′ ~D ′ �o>U

y²: w,µ∇ · ~D ′ = ∇ · ~D = ρ =

{ρ1 0� 1

ρ2 0� 2, ~E = −∇ϕ, ~E ′ = −∇ϕ′,∮

S1

σ1 ds =

∮S1

σ′1 ds = Q1,

∮S2

σ2 ds =

∮S2

σ′2 ds = Q2 d?± ds L«¡È©

o·>UµU ′ − U =1

2

∫ε ~E

′ · ~E′dτ −

1

2

∫ε ~E · ~Edτ

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

o�±ò Q1 Ú Q2 ©Ù¤ σ1 Ú σ2§¦�0�L¡��³¡§ù��m>|�µ ~E, ~D

Thomson ½n �ѵ| ~E, ~D �o>U�u| ~E ′ ~D ′ �o>U

y²: w,µ∇ · ~D ′ = ∇ · ~D = ρ =

{ρ1 0� 1

ρ2 0� 2, ~E = −∇ϕ, ~E ′ = −∇ϕ′,∮

S1

σ1 ds =

∮S1

σ′1 ds = Q1,

∮S2

σ2 ds =

∮S2

σ′2 ds = Q2 d?± ds L«¡È©

o·>UµU ′ − U =1

2

∫ε ~E

′ · ~E′dτ −

1

2

∫ε ~E · ~Edτ

=1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫ε ~E · ( ~E

′ − ~E)dτ︸ ︷︷ ︸A =

∫~E · (ε ~E

′ − ε ~E)dτ

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

n!Thomson ½nThomson ½nµ 3�½u�50�¥��NX§���Nþ�>Ö©Ù

¦���Nþ��³N�§NXo>U4�§NX��·>²ï"

�lÔnþn)T½n§�Äü�0���¹µ0� 1 Ú0� 2 þ©OkNgd>Ö©Ù ρ1 Ú ρ2

y3�\ Q1 Ú Q2 >þ§�©O©Ù�0� 1 Ú0� 2 L¡"

b�\ò Q1 Ú Q2 ©Ù¤ σ′1 Ú σ′2§éAu©Ùµρ1, ρ2, σ′1, σ′2§�m>|�µ ~E ′, ~D ′

o�±ò Q1 Ú Q2 ©Ù¤ σ1 Ú σ2§¦�0�L¡��³¡§ù��m>|�µ ~E, ~D

Thomson ½n �ѵ| ~E, ~D �o>U�u| ~E ′ ~D ′ �o>U

y²: w,µ∇ · ~D ′ = ∇ · ~D = ρ =

{ρ1 0� 1

ρ2 0� 2, ~E = −∇ϕ, ~E ′ = −∇ϕ′,∮

S1

σ1 ds =

∮S1

σ′1 ds = Q1,

∮S2

σ2 ds =

∮S2

σ′2 ds = Q2 d?± ds L«¡È©

o·>UµU ′ − U =1

2

∫ε ~E

′ · ~E′dτ −

1

2

∫ε ~E · ~Edτ

=1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫ε ~E · ( ~E

′ − ~E)dτ︸ ︷︷ ︸A =

∫~E · (ε ~E

′ − ε ~E)dτ

E��Æ ÔnX ��� Mï� 7

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ,

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ |^ (∇ϕ) · ~a = ∇ · (ϕ~a)− ϕ∇ · ~a

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ |^ (∇ϕ) · ~a = ∇ · (ϕ~a)− ϕ∇ · ~a

= −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ +

∫ϕ∇ · ( ~D

′ − ~D)︸ ︷︷ ︸0

dτ = −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ |^ (∇ϕ) · ~a = ∇ · (ϕ~a)− ϕ∇ · ~a

= −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ +

∫ϕ∇ · ( ~D

′ − ~D)︸ ︷︷ ︸0

dτ = −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ e¡y² A = 0

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ |^ (∇ϕ) · ~a = ∇ · (ϕ~a)− ϕ∇ · ~a

= −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ +

∫ϕ∇ · ( ~D

′ − ~D)︸ ︷︷ ︸0

dτ = −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ e¡y² A = 0

�mk 3 �þ!©« V1, V2 Ú V3§X㤫"

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ |^ (∇ϕ) · ~a = ∇ · (ϕ~a)− ϕ∇ · ~a

= −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ +

∫ϕ∇ · ( ~D

′ − ~D)︸ ︷︷ ︸0

dτ = −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ e¡y² A = 0

�mk 3 �þ!©« V1, V2 Ú V3§X㤫"

A = −∫

V1+V2+V3

∇ ·[ϕ( ~D

′ − ~D)]dτ .¡ü>�ȼêØëY§�©¤ 3 �«È©

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ |^ (∇ϕ) · ~a = ∇ · (ϕ~a)− ϕ∇ · ~a

= −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ +

∫ϕ∇ · ( ~D

′ − ~D)︸ ︷︷ ︸0

dτ = −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ e¡y² A = 0

�mk 3 �þ!©« V1, V2 Ú V3§X㤫"

A = −∫

V1+V2+V3

∇ ·[ϕ( ~D

′ − ~D)]dτ .¡ü>�ȼêØëY§�©¤ 3 �«È©

= −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds V1 �S.¡©O� S1i§S1e

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds−

∮S∞

~n ·[ϕ( ~D

′ − ~D)]ds︸ ︷︷ ︸

é·|§Ã¡�¡�È©� 0

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ |^ (∇ϕ) · ~a = ∇ · (ϕ~a)− ϕ∇ · ~a

= −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ +

∫ϕ∇ · ( ~D

′ − ~D)︸ ︷︷ ︸0

dτ = −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ e¡y² A = 0

�mk 3 �þ!©« V1, V2 Ú V3§X㤫"

A = −∫

V1+V2+V3

∇ ·[ϕ( ~D

′ − ~D)]dτ .¡ü>�ȼêØëY§�©¤ 3 �«È©

= −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds V1 �S.¡©O� S1i§S1e

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds−

∮S∞

~n ·[ϕ( ~D

′ − ~D)]ds︸ ︷︷ ︸

é·|§Ã¡�¡�È©� 0V2 �S.¡©O� S2i§S2e§

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ |^ (∇ϕ) · ~a = ∇ · (ϕ~a)− ϕ∇ · ~a

= −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ +

∫ϕ∇ · ( ~D

′ − ~D)︸ ︷︷ ︸0

dτ = −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ e¡y² A = 0

�mk 3 �þ!©« V1, V2 Ú V3§X㤫"

A = −∫

V1+V2+V3

∇ ·[ϕ( ~D

′ − ~D)]dτ .¡ü>�ȼêØëY§�©¤ 3 �«È©

= −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds V1 �S.¡©O� S1i§S1e

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds−

∮S∞

~n ·[ϕ( ~D

′ − ~D)]ds︸ ︷︷ ︸

é·|§Ã¡�¡�È©� 0V2 �S.¡©O� S2i§S2e§ S1i = −S1e§S2i = −S2e£����¤

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

U ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E)dτ︸ ︷︷ ︸A

A =

∫~E · ( ~D

′ − ~D)dτ, Ù¥∇ · ( ~D′ − ~D) = 0

= −∫

(∇ϕ) · ( ~D′ − ~D)dτ |^ (∇ϕ) · ~a = ∇ · (ϕ~a)− ϕ∇ · ~a

= −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ +

∫ϕ∇ · ( ~D

′ − ~D)︸ ︷︷ ︸0

dτ = −∫∇ ·

[ϕ( ~D

′ − ~D)]dτ e¡y² A = 0

�mk 3 �þ!©« V1, V2 Ú V3§X㤫"

A = −∫

V1+V2+V3

∇ ·[ϕ( ~D

′ − ~D)]dτ .¡ü>�ȼêØëY§�©¤ 3 �«È©

= −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds V1 �S.¡©O� S1i§S1e

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds−

∮S∞

~n ·[ϕ( ~D

′ − ~D)]ds︸ ︷︷ ︸

é·|§Ã¡�¡�È©� 0V2 �S.¡©O� S2i§S2e§ S1i = −S1e§S2i = −S2e£����¤

c®b½ ϕ éAu0�L¡��³¡�)§�.¡þ ϕ1i = ϕ1e§ϕ2i = ϕ2e

Ï §A ��¤�

E��Æ ÔnX ��� Mï� 8

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

Ù¥|^µ30�L¡ϕ1i = ϕ1e = ϕ1 = ~êϕ2i = ϕ2e = ϕ2 = ~ê

�µS1i = −S1e = S1

S2i = −S2e = S2

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

Ù¥|^µ30�L¡ϕ1i = ϕ1e = ϕ1 = ~êϕ2i = ϕ2e = ϕ2 = ~ê

�µS1i = −S1e = S1

S2i = −S2e = S2

= −ϕ1

∮S1

σ′1 ds︸ ︷︷ ︸

Q1

+ ϕ1

∮S1

σ1 ds︸ ︷︷ ︸Q1

− ϕ2

∮S2

σ′2 ds︸ ︷︷ ︸

Q2

+ ϕ2

∮S2

σ2 ds︸ ︷︷ ︸Q2

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

Ù¥|^µ30�L¡ϕ1i = ϕ1e = ϕ1 = ~êϕ2i = ϕ2e = ϕ2 = ~ê

�µS1i = −S1e = S1

S2i = −S2e = S2

= −ϕ1

∮S1

σ′1 ds︸ ︷︷ ︸

Q1

+ ϕ1

∮S1

σ1 ds︸ ︷︷ ︸Q1

− ϕ2

∮S2

σ′2 ds︸ ︷︷ ︸

Q2

+ ϕ2

∮S2

σ2 ds︸ ︷︷ ︸Q2

= 0

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

Ù¥|^µ30�L¡ϕ1i = ϕ1e = ϕ1 = ~êϕ2i = ϕ2e = ϕ2 = ~ê

�µS1i = −S1e = S1

S2i = −S2e = S2

= −ϕ1

∮S1

σ′1 ds︸ ︷︷ ︸

Q1

+ ϕ1

∮S1

σ1 ds︸ ︷︷ ︸Q1

− ϕ2

∮S2

σ′2 ds︸ ︷︷ ︸

Q2

+ ϕ2

∮S2

σ2 ds︸ ︷︷ ︸Q2

= 0

�µU ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E) dτ︸ ︷︷ ︸A = 0

≥ 0 =⇒ U ′ ≥ U

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

Ù¥|^µ30�L¡ϕ1i = ϕ1e = ϕ1 = ~êϕ2i = ϕ2e = ϕ2 = ~ê

�µS1i = −S1e = S1

S2i = −S2e = S2

= −ϕ1

∮S1

σ′1 ds︸ ︷︷ ︸

Q1

+ ϕ1

∮S1

σ1 ds︸ ︷︷ ︸Q1

− ϕ2

∮S2

σ′2 ds︸ ︷︷ ︸

Q2

+ ϕ2

∮S2

σ2 ds︸ ︷︷ ︸Q2

= 0

�µU ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E) dτ︸ ︷︷ ︸A = 0

≥ 0 =⇒ U ′ ≥ U

é0�§�½�Щ¡>Ö©Ù=¦ØU¦·>U4�£éAu0�L¡��³¡¤§�>Ö

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

Ù¥|^µ30�L¡ϕ1i = ϕ1e = ϕ1 = ~êϕ2i = ϕ2e = ϕ2 = ~ê

�µS1i = −S1e = S1

S2i = −S2e = S2

= −ϕ1

∮S1

σ′1 ds︸ ︷︷ ︸

Q1

+ ϕ1

∮S1

σ1 ds︸ ︷︷ ︸Q1

− ϕ2

∮S2

σ′2 ds︸ ︷︷ ︸

Q2

+ ϕ2

∮S2

σ2 ds︸ ︷︷ ︸Q2

= 0

�µU ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E) dτ︸ ︷︷ ︸A = 0

≥ 0 =⇒ U ′ ≥ U

é0�§�½�Щ¡>Ö©Ù=¦ØU¦·>U4�£éAu0�L¡��³¡¤§�>Ö3ý�0�þØUgd£Ä§���±�½�¡>Ö©Ù" �â?�§â¡�±Ø²"

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

Ù¥|^µ30�L¡ϕ1i = ϕ1e = ϕ1 = ~êϕ2i = ϕ2e = ϕ2 = ~ê

�µS1i = −S1e = S1

S2i = −S2e = S2

= −ϕ1

∮S1

σ′1 ds︸ ︷︷ ︸

Q1

+ ϕ1

∮S1

σ1 ds︸ ︷︷ ︸Q1

− ϕ2

∮S2

σ′2 ds︸ ︷︷ ︸

Q2

+ ϕ2

∮S2

σ2 ds︸ ︷︷ ︸Q2

= 0

�µU ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E) dτ︸ ︷︷ ︸A = 0

≥ 0 =⇒ U ′ ≥ U

é0�§�½�Щ¡>Ö©Ù=¦ØU¦·>U4�£éAu0�L¡��³¡¤§�>Ö3ý�0�þØUgd£Ä§���±�½�¡>Ö©Ù" �â?�§â¡�±Ø²"

é�N§du>ÖUgd£Ä§=¦<��½Ð©¡>Ö©Ù§>Ö�ògÄ­#©Ù§

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

Ù¥|^µ30�L¡ϕ1i = ϕ1e = ϕ1 = ~êϕ2i = ϕ2e = ϕ2 = ~ê

�µS1i = −S1e = S1

S2i = −S2e = S2

= −ϕ1

∮S1

σ′1 ds︸ ︷︷ ︸

Q1

+ ϕ1

∮S1

σ1 ds︸ ︷︷ ︸Q1

− ϕ2

∮S2

σ′2 ds︸ ︷︷ ︸

Q2

+ ϕ2

∮S2

σ2 ds︸ ︷︷ ︸Q2

= 0

�µU ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E) dτ︸ ︷︷ ︸A = 0

≥ 0 =⇒ U ′ ≥ U

é0�§�½�Щ¡>Ö©Ù=¦ØU¦·>U4�£éAu0�L¡��³¡¤§�>Ö3ý�0�þØUgd£Ä§���±�½�¡>Ö©Ù" �â?�§â¡�±Ø²"

é�N§du>ÖUgd£Ä§=¦<��½Ð©¡>Ö©Ù§>Ö�ògÄ­#©Ù§¦�·>U4�§�Ò´¦��NL¡��³¡" �Y?�§Y¡�½²"

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

A = −∮

S1i

~n ·[ϕ1i( ~D

′1i − ~D1i)

]ds−

∮S1e

~n ·[ϕ1e( ~D

′1e − ~D1e)

]ds

−∮

S2i

~n ·[ϕ2i( ~D

′2i − ~D2i)

]ds−

∮S2e

~n ·[ϕ2e( ~D

′2e − ~D2e)

]ds

= ϕ1

∮S1

~n · ( ~D′1e − ~D

′1i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′1

ds− ϕ1

∮S1

~n · ( ~D1e − ~D1i)︸ ︷︷ ︸.¡>Ö©Ù−σ1

ds

+ ϕ2

∮S2

~n · ( ~D′2e − ~D

′2i)︸ ︷︷ ︸

.¡>Ö©Ù−σ′2

ds− ϕ2

∮S2

~n · ( ~D2e − ~D2i)︸ ︷︷ ︸.¡>Ö©Ù−σ2

ds

Ù¥|^µ30�L¡ϕ1i = ϕ1e = ϕ1 = ~êϕ2i = ϕ2e = ϕ2 = ~ê

�µS1i = −S1e = S1

S2i = −S2e = S2

= −ϕ1

∮S1

σ′1 ds︸ ︷︷ ︸

Q1

+ ϕ1

∮S1

σ1 ds︸ ︷︷ ︸Q1

− ϕ2

∮S2

σ′2 ds︸ ︷︷ ︸

Q2

+ ϕ2

∮S2

σ2 ds︸ ︷︷ ︸Q2

= 0

�µU ′ − U =1

2

∫ε( ~E

′ − ~E)2dτ︸ ︷︷ ︸

·|¥ ε > 0§�d�≥ 0

+

∫~E · (ε ~E

′ − ε ~E) dτ︸ ︷︷ ︸A = 0

≥ 0 =⇒ U ′ ≥ U

é0�§�½�Щ¡>Ö©Ù=¦ØU¦·>U4�£éAu0�L¡��³¡¤§�>Ö3ý�0�þØUgd£Ä§���±�½�¡>Ö©Ù" �â?�§â¡�±Ø²"

é�N§du>ÖUgd£Ä§=¦<��½Ð©¡>Ö©Ù§>Ö�ògÄ­#©Ù§¦�·>U4�§�Ò´¦��NL¡��³¡" �Y?�§Y¡�½²"

E��Æ ÔnX ��� Mï� 9

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

I =

∫∇ · (ϕ∇ϕ

′) dτ −

∫ϕ(∇2

ϕ′) dτ

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

I =

∫∇ · (ϕ∇ϕ

′) dτ −

∫ϕ(∇2

ϕ′) dτ |^∇2

ϕ′= −

1

ε0

ρ′

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

I =

∫∇ · (ϕ∇ϕ

′) dτ −

∫ϕ(∇2

ϕ′) dτ |^∇2

ϕ′= −

1

ε0

ρ′

=

∮~n · [ϕ∇ϕ

′] dσ +

1

ε0

∫ϕρ

′dτ

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

I =

∫∇ · (ϕ∇ϕ

′) dτ −

∫ϕ(∇2

ϕ′) dτ |^∇2

ϕ′= −

1

ε0

ρ′

=

∮~n · [ϕ∇ϕ

′] dσ +

1

ε0

∫ϕρ

′dτ 1���á�?¡È©§� 0

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

I =

∫∇ · (ϕ∇ϕ

′) dτ −

∫ϕ(∇2

ϕ′) dτ |^∇2

ϕ′= −

1

ε0

ρ′

=

∮~n · [ϕ∇ϕ

′] dσ +

1

ε0

∫ϕρ

′dτ 1���á�?¡È©§� 0

=1

ε0

∫ϕρ

′dτ

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

I =

∫∇ · (ϕ∇ϕ

′) dτ −

∫ϕ(∇2

ϕ′) dτ |^∇2

ϕ′= −

1

ε0

ρ′

=

∮~n · [ϕ∇ϕ

′] dσ +

1

ε0

∫ϕρ

′dτ 1���á�?¡È©§� 0

=1

ε0

∫ϕρ

′dτ =

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

I =

∫∇ · (ϕ∇ϕ

′) dτ −

∫ϕ(∇2

ϕ′) dτ |^∇2

ϕ′= −

1

ε0

ρ′

=

∮~n · [ϕ∇ϕ

′] dσ +

1

ε0

∫ϕρ

′dτ 1���á�?¡È©§� 0

=1

ε0

∫ϕρ

′dτ =⇑

Ón

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

I =

∫∇ · (ϕ∇ϕ

′) dτ −

∫ϕ(∇2

ϕ′) dτ |^∇2

ϕ′= −

1

ε0

ρ′

=

∮~n · [ϕ∇ϕ

′] dσ +

1

ε0

∫ϕρ

′dτ 1���á�?¡È©§� 0

=1

ε0

∫ϕρ

′dτ =

1

ε0

∫ϕ′ρ dτ

⇑Ón

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

íص l>ÖXÚé��Ø�>�N¬É>ÖXÚáÚ£�á p64 ~ 1¤"

o!��p´½n (Green’s reciprocity theorem)

��p´½nµ ��mk�>Ö©Ù ρ(~r)§3�m�)�>³� ϕ(~r)§XJrd>Ö©Ù�¤ ρ′(~r)§3�m�>³�A/C� ϕ′(~r)§kµ∫

V∞

ρ(~r) ϕ′(~r) dτ =∫

V∞

ρ′(~r) ϕ(~r) dτ

y²µ

- I =

∫(∇ϕ) · (∇ϕ

′) dτ

|^µ ∇ · (ϕ∇ϕ′) = (∇ϕ) · (∇ϕ′) + ϕ(∇2ϕ′)

I =

∫∇ · (ϕ∇ϕ

′) dτ −

∫ϕ(∇2

ϕ′) dτ |^∇2

ϕ′= −

1

ε0

ρ′

=

∮~n · [ϕ∇ϕ

′] dσ +

1

ε0

∫ϕρ

′dτ 1���á�?¡È©§� 0

=1

ε0

∫ϕρ

′dτ =

1

ε0

∫ϕ′ρ dτ �y

⇑Ón

E��Æ ÔnX ��� Mï� 10

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

=1

ε0

∑i

ϕi Q′i

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

=1

ε0

∑i

ϕi Q′i Q′

i ��N i éAu1�«>Ö©Ù�>þ

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

=1

ε0

∑i

ϕi Q′i Q′

i ��N i éAu1�«>Ö©Ù�>þ

=1

ε0

∑i

ϕ′i Qi

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

=1

ε0

∑i

ϕi Q′i Q′

i ��N i éAu1�«>Ö©Ù�>þ

=1

ε0

∑i

ϕ′i Qi Qi ��N i éAu1�«>Ö©Ù�>þ

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

=1

ε0

∑i

ϕi Q′i Q′

i ��N i éAu1�«>Ö©Ù�>þ

=1

ε0

∑i

ϕ′i Qi Qi ��N i éAu1�«>Ö©Ù�>þ

∑i

ϕ′i Qi =

∑i

ϕi Q′i

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

=1

ε0

∑i

ϕi Q′i Q′

i ��N i éAu1�«>Ö©Ù�>þ

=1

ε0

∑i

ϕ′i Qi Qi ��N i éAu1�«>Ö©Ù�>þ

∑i

ϕ′i Qi =

∑i

ϕi Q′i

ϕiµ

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

=1

ε0

∑i

ϕi Q′i Q′

i ��N i éAu1�«>Ö©Ù�>þ

=1

ε0

∑i

ϕ′i Qi Qi ��N i éAu1�«>Ö©Ù�>þ

∑i

ϕ′i Qi =

∑i

ϕi Q′i

ϕiµ éAu1�«>Ö©Ù§=�N k �> Qk � (k = 1, 2, . . . , n)§1 i ��Nþ�>³

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

=1

ε0

∑i

ϕi Q′i Q′

i ��N i éAu1�«>Ö©Ù�>þ

=1

ε0

∑i

ϕ′i Qi Qi ��N i éAu1�«>Ö©Ù�>þ

∑i

ϕ′i Qi =

∑i

ϕi Q′i

ϕiµ éAu1�«>Ö©Ù§=�N k �> Qk � (k = 1, 2, . . . , n)§1 i ��Nþ�>³

ϕ′iµ

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

é�NXµ

I =1

ε0

∫ϕρ

′dτ é��N i ¦Ú§¿�Ä��N>Ö©Ù3L¡

=1

ε0

∑i

∮Si

ϕi σ′qi dσi σ′qi ��N i �1�«¡>Ö�Ý

=1

ε0

∑i

ϕi

∮Si

σ′qi dσi 1�«>Ö©Ù3�N i þ�)�³ ϕi �~ê

=1

ε0

∑i

ϕi Q′i Q′

i ��N i éAu1�«>Ö©Ù�>þ

=1

ε0

∑i

ϕ′i Qi Qi ��N i éAu1�«>Ö©Ù�>þ

∑i

ϕ′i Qi =

∑i

ϕi Q′i

ϕiµ éAu1�«>Ö©Ù§=�N k �> Qk � (k = 1, 2, . . . , n)§1 i ��Nþ�>³

ϕ′iµ éAu1�«>Ö©Ù§=�N k �> Q′k � (k = 1, 2, . . . , n)§1 i ��Nþ�>³

E��Æ ÔnX ��� Mï� 11

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

1�«©Ù§�k�N j �>³ ϕ′j 6= 0

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

1�«©Ù§�k�N j �>³ ϕ′j 6= 0

Ïdµ ϕiQ′i =

∑k

ϕkQ′k =

∑k

ϕ′kQk = ϕ′jQj

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

1�«©Ù§�k�N j �>³ ϕ′j 6= 0

Ïdµ ϕiQ′i =

∑k

ϕkQ′k =

∑k

ϕ′kQk = ϕ′jQj

⇓

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

1�«©Ù§�k�N j �>³ ϕ′j 6= 0

Ïdµ ϕiQ′i =

∑k

ϕkQ′k =

∑k

ϕ′kQk = ϕ′jQj

⇓

ϕiCijϕ′j = ϕ′jCjiϕi

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

1�«©Ù§�k�N j �>³ ϕ′j 6= 0

Ïdµ ϕiQ′i =

∑k

ϕkQ′k =

∑k

ϕ′kQk = ϕ′jQj

⇓

ϕiCijϕ′j = ϕ′jCjiϕi =⇒ Cij = Cji

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

1�«©Ù§�k�N j �>³ ϕ′j 6= 0

Ïdµ ϕiQ′i =

∑k

ϕkQ′k =

∑k

ϕ′kQk = ϕ′jQj

⇓

ϕiCijϕ′j = ϕ′jCjiϕi =⇒ Cij = Cji

~ 3µ�á p70, ~K5

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

1�«©Ù§�k�N j �>³ ϕ′j 6= 0

Ïdµ ϕiQ′i =

∑k

ϕkQ′k =

∑k

ϕ′kQk = ϕ′jQj

⇓

ϕiCijϕ′j = ϕ′jCjiϕi =⇒ Cij = Cji

~ 3µ�á p70, ~K5

† “�/”´ÚåÔnþ�ÜÂ"Ï��/3,«¿Âþ´�±�´�N� �UC§l ¦<ر� Cij �u)UC

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

1�«©Ù§�k�N j �>³ ϕ′j 6= 0

Ïdµ ϕiQ′i =

∑k

ϕkQ′k =

∑k

ϕ′kQk = ϕ′jQj

⇓

ϕiCijϕ′j = ϕ′jCjiϕi =⇒ Cij = Cji

~ 3µ�á p70, ~K5

† “�/”´ÚåÔnþ�ÜÂ"Ï��/3,«¿Âþ´�±�´�N� �UC§l ¦<ر� Cij �u)UC

¢Sþ§�±@�<�3��N¥þ��>Ö§¦�1 i ��N>³� ϕi§Ù{�N>³� 0

E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

~ 2µ|^��p´½ny²>aXê÷v Cij = Cji"k n ��N�¤��NX

1�«>Ö>³©Ù�µ 1 i ��N>³� ϕi§Ù{�N�/§† ù�1 j ��N�> Qj

1�«>Ö>³©Ù�µ 1 j ��N>³� ϕ′j§Ù{�N�/§† ù�1 i ��N�> Q′i

d>aXê�½Âµ Qj =∑

k

Cjkϕk = Cjiϕi, éAu1�«>Ö>³©Ù

Q′i =

∑k

Cikϕ′k = Cijϕ

′j, éAu1�«>Ö>³©Ù

d��p´½nµ∑

k

ϕkQ′k =

∑k

ϕ′kQk 1�«©Ù§�k�N i �>³ ϕi 6= 0

1�«©Ù§�k�N j �>³ ϕ′j 6= 0

Ïdµ ϕiQ′i =

∑k

ϕkQ′k =

∑k

ϕ′kQk = ϕ′jQj

⇓

ϕiCijϕ′j = ϕ′jCjiϕi =⇒ Cij = Cji

~ 3µ�á p70, ~K5

† “�/”´ÚåÔnþ�ÜÂ"Ï��/3,«¿Âþ´�±�´�N� �UC§l ¦<ر� Cij �u)UC

¢Sþ§�±@�<�3��N¥þ��>Ö§¦�1 i ��N>³� ϕi§Ù{�N>³� 0

@o§;�X�¯K´§UÄÏL��>Ö¢yù«>³©Ùº g�E��Æ ÔnX ��� Mï� 12

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

�NS ~E1 = 0, ~D1 = 0 ~Fs = (~n · ~D2) ~E2 −12( ~D2 · ~E2)~n

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

�NS ~E1 = 0, ~D1 = 0 ~Fs = (~n · ~D2) ~E2 −12( ~D2 · ~E2)~n S1 Ú Sc �¡È©� 0

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

�NS ~E1 = 0, ~D1 = 0 ~Fs = (~n · ~D2) ~E2 −12( ~D2 · ~E2)~n S1 Ú Sc �¡È©� 0

>|÷�NL¡{�µ ~Fs =ε

2~E2

2 ~n

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

�NS ~E1 = 0, ~D1 = 0 ~Fs = (~n · ~D2) ~E2 −12( ~D2 · ~E2)~n S1 Ú Sc �¡È©� 0

>|÷�NL¡{�µ ~Fs =ε

2~E2

2 ~n £b��N��5��Ó50�¤

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

�NS ~E1 = 0, ~D1 = 0 ~Fs = (~n · ~D2) ~E2 −12( ~D2 · ~E2)~n S1 Ú Sc �¡È©� 0

>|÷�NL¡{�µ ~Fs =ε

2~E2

2 ~n £b��N��5��Ó50�¤

�NS ~E1 = 0, ~D1 = 0µ ~n · ( ~D2 − ~D1) = σq

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

�NS ~E1 = 0, ~D1 = 0 ~Fs = (~n · ~D2) ~E2 −12( ~D2 · ~E2)~n S1 Ú Sc �¡È©� 0

>|÷�NL¡{�µ ~Fs =ε

2~E2

2 ~n £b��N��5��Ó50�¤

�NS ~E1 = 0, ~D1 = 0µ ~n · ( ~D2 − ~D1) = σq⇒ ~n · (εE2~n) = σq

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

�NS ~E1 = 0, ~D1 = 0 ~Fs = (~n · ~D2) ~E2 −12( ~D2 · ~E2)~n S1 Ú Sc �¡È©� 0

>|÷�NL¡{�µ ~Fs =ε

2~E2

2 ~n £b��N��5��Ó50�¤

�NS ~E1 = 0, ~D1 = 0µ ~n · ( ~D2 − ~D1) = σq⇒ ~n · (εE2~n) = σq⇒ ~E2 =σq

ε~n

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

�NS ~E1 = 0, ~D1 = 0 ~Fs = (~n · ~D2) ~E2 −12( ~D2 · ~E2)~n S1 Ú Sc �¡È©� 0

>|÷�NL¡{�µ ~Fs =ε

2~E2

2 ~n £b��N��5��Ó50�¤

�NS ~E1 = 0, ~D1 = 0µ ~n · ( ~D2 − ~D1) = σq⇒ ~n · (εE2~n) = σq⇒ ~E2 =σq

ε~n

�Nü ¡ÈÉåµ ~Fs =σ2

q

2ε~n KØå

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

Ê!�^u�NL¡�·>å

�3·>|¥��N¬É�·>|��^å"

�^u�NL¡ dσ �uü �m6\�N¡�Äþ

Äþ6�ÝÜþµ↔T =

12( ~D · ~E)

↔I − ~D ~E

�NL¡Éåµ ~Fs = −∮S(~n ·

↔T )dσ

S ´�A²Ý§�¹µ �NSÜ�¡ S1§R�uL¡�ý¡ ScÚ�NÜ�¡ S2

�NS ~E1 = 0, ~D1 = 0 ~Fs = (~n · ~D2) ~E2 −12( ~D2 · ~E2)~n S1 Ú Sc �¡È©� 0

>|÷�NL¡{�µ ~Fs =ε

2~E2

2 ~n £b��N��5��Ó50�¤

�NS ~E1 = 0, ~D1 = 0µ ~n · ( ~D2 − ~D1) = σq⇒ ~n · (εE2~n) = σq⇒ ~E2 =σq

ε~n

�Nü ¡ÈÉåµ ~Fs =σ2

q

2ε~n KØå

���NÉåµ ~F =∮

σ2q

2ε~n dσ

E��Æ ÔnX ��� Mï� 13

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

g�µ

E��Æ ÔnX ��� Mï� 14

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

g�µ ·>|>^Äþ�Ý ~g = ε0 ~E × ~B = 0

E��Æ ÔnX ��� Mï� 14

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

g�µ ·>|>^Äþ�Ý ~g = ε0 ~E × ~B = 0

O�Éå�§Äþ6�Ý↔T 6= 0

E��Æ ÔnX ��� Mï� 14

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

g�µ ·>|>^Äþ�Ý ~g = ε0 ~E × ~B = 0

O�Éå�§Äþ6�Ý↔T 6= 0

⇒ vkÄþ�ݧÛ5Äþ6?

E��Æ ÔnX ��� Mï� 14

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

g�µ ·>|>^Äþ�Ý ~g = ε0 ~E × ~B = 0

O�Éå�§Äþ6�Ý↔T 6= 0

⇒ vkÄþ�ݧÛ5Äþ6?

Uþ�Ý!Äþ�Ý!Äþ6�Ý£n�¤Üþ�¤�éØ¥�o�UÄÜþ§´��Ônþ"ù�Ônþ7L^o�UÄÜþ£� 16 �©þ¤âU��£ã"vkn�Äþ§��uo�UÄÜþ¥,©þ� 0§�n�Äþ6Üþ´o�UÄÜþ�Ù§©þ§�±Ø� 0"

E��Æ ÔnX ��� Mï� 14

Let there be light²;>ÄåÆ�Ø

1nÙµ·>| § 3.3

g�µ ·>|>^Äþ�Ý ~g = ε0 ~E × ~B = 0

O�Éå�§Äþ6�Ý↔T 6= 0

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