B-field of a rotating charged conducting sphere 1

Magnetic Field of a Rotating Charged Conducting Sphere

© Frits F.M. de Mul

B-field of a rotating charged conducting sphere 2

B-field of a rotating charged conducting sphere

Question:

Calculate B-field in arbitrary points on the axis of rotation inside and outside the sphere

Available:

A charged conducting sphere (charge Q, radius R), rotating with ω rad/sec

ω

B-field of a rotating charged conducting sphere 3

Calculate B-field in point P inside or outside the sphere

P

P

O

ω

Analysis and Symmetry (1)

Assume Z-axis through O and P.

zP

Z

Y

XCoordinate systems:

- X,Y, Z

θ

ϕ

r

- r, θ, ϕ

B-field of a rotating charged conducting sphere 4

Analysis and Symmetry (2)

Conducting sphere,

all charges at surface:

surface density: σ = Q/(4πR2) [C/m2]

ω

P

P

zP

Y

X

Z

θ

ϕ

r

ORotating charges will establish a “surface current”

Surface current density j’ [A/m] will be a function of θ

j’

B-field of a rotating charged conducting sphere 5

Analysis and Symmetry (3)

ω

P

zP

Y

X

Z

θ

ϕ

r

O

T

Cylinder- symmetry around Z-axis:

dBz

Z-components only !!

Direction of contributions dB:

P

O

dB

T

θ

r

er

dl

20 .

4 Pr

I redldB

×=π

µBiot & Savart :

rPdB dB, dl and er mutual. perpendic.

B-field of a rotating charged conducting sphere 6

Approach (1): a long wire

dB

20 .

4 Pr

I redldB

×=π

µBiot & Savart :

note:

r and vector er !!

dB ⊥ dl and er

dB ⊥ ∆ AOP

Z

YX

P

z

I.dl in dz at z

dl

er rP

yP

α

A

O

B-field of a rotating charged conducting sphere 7

Approach (2): a volume current

dB

dvrP

20

4rej

dB×=

πµ

Biot & Savart :

dB ⊥ dl and er

dB ⊥ ∆ AOP

j: current density [A/m2]

Z

Y

P

j.dA.dl = j.dvdl

er

yP

dA

jA

OrP

B-field of a rotating charged conducting sphere 8

Approach (3): a surface current

dB

dArP

20

4rej

dB×′

=π

µ

Biot & Savart :

dB ⊥ dl and er

dB ⊥ ∆ AOP

Z

Y

P

dl

er

yP

dl

j’A

OrP

Current strip at surface: j’: current density[A/m]

j’.db.dl = j’.dA

dldb

B-field of a rotating charged conducting sphere 9

Approach (4)

Z

dϕ

R

ϕ

θ

dθ

R sinθ

Conducting sphere,

surface density: σ = Q/(4πR2)

surface element:

dA = (R.dθ).(R.sinθ. dϕ)

R.dθ.R.sinθ. dϕ

Surface element:

B-field of a rotating charged conducting sphere 10

Conducting sphere (1)

dA = db.dl

Surface charge σ.dA on dA will rotate with ω

dl = R.sinθ.dϕ ; db= R dθ Needed:

• j , er , rP

dArP

20

4rej

dB×′=

πµ with j’ in [A/m]

R.sinθ.dϕ

Z

R

ϕ

dθ

dϕ

R sinθ R.dθ

ω

θ

B-field of a rotating charged conducting sphere 11

Conducting sphere (2)

Z

R

ϕ

dθ

dϕ

R sinθ

R.dθ R.sinθ.dϕ

ω

dA = db.dl

dl = R.sinθ.dϕ db= Rdθ

Full rotation over 2π.Rsinθ in 2π/ω s.

Charge on ring with radius R.sinθ and width db is: σ. 2πR.sinθ . db

current: dI = σ.2πR.sinθ.db / (2π/ω) = σ ω R sinθ .db

current density: j’ =σωR sinθ [A/m]

θ

B-field of a rotating charged conducting sphere 12

Conducting sphere (3)

R

ϕ

dθ

dϕ

R sinθ

R.dθ R.sinθ.dϕ

dArP

20

4rej

dB×′

=π

µP

zP

j’

errP

dA = R.dθ. R.sinθ. dϕ

j’ ⊥ er :

=> | j’ x er | = j’.er = j’

j’ = σωR sin θ

ω

θ

B-field of a rotating charged conducting sphere 13

Conducting sphere (4)

R

ϕ

dθ

dϕ

R sinθ

P

zP

j’

errP

dArP

20

4rej

dB×′

=π

µ

dA = Rdθ R.sinθ. dϕ

Z-components only !!

dBz α

Pr

R θα sincos =

Cylinder- symmetry: P

O

dB

θ R

rPzP α er

j’ = σωR sin θ

ω

θ

B-field of a rotating charged conducting sphere 14

Conducting sphere (5)

R

ϕ

dθ

dϕ

R sinθ

P

zP

j’

errP

dArP

20

4rej

dB×′

=π

µ

dA = Rdθ.R.sinθ. dϕ

P

O

dBdBz

θ

α

R

rPzP α

Pr

R θα sincos =

rP2= (R.sinθ)2 +

(zP - R.cosθ)2

PP

z r

RdRdR

r

RdB

θϕθθθσωπ

µ sinsin...

sin

4 20=

j’ = σωR sin θ

ω

θ

B-field of a rotating charged conducting sphere 15

Conducting sphere (6)

R

ϕ

dθ

dϕ

R sinθ

P

zP

j’

errP

with rP2= (R.sinθ)2 + (zP - R.cosθ)2

ϕθθσωπ

µdd

r

RdB

P

z .sin

4 3

340=

Integration: 0<θ<π ; 0<ϕ< 2π

PP

z r

RdRdR

r

RdB

θϕθθθσωπ

µ sinsin...

sin

4 20=

ω

θ

B-field of a rotating charged conducting sphere 16

Conducting sphere (7)

ω

P

P

zP

Y

X

Z

θ

ϕ

R

O

this result holds for zP>R ;

for -R<zP<R the result is:

zeB Rσωµ03

2 =

and for zP<-R: zeB 3

40

.3

2

pz

R

−= σωµ

zeB 3

40

.3

2 :result

pz

Rσωµ=

B-field of a rotating charged conducting sphere 17

Conducting sphere (8)

inside sphere: constant field !!ω

P

P

zP

Y

X

Z

θ

ϕ

r

O

result for |zP|>R :

result for |zP|<R :

zeB 3

40

.3

2

pz

Rσωµ=

zeB Rσωµ03

2 =

B directed along +ez for all points everywhere on Z-axis !!

B-field of a rotating charged conducting sphere 18

Conducting sphere (9)

With surface density: σ = Q/(4πR2) :

result for |zP| > R : zz eeB 3

20

3

40

6.3

2

pp z

RQ

z

R

πωµσωµ ==

result for |zP| < R : zz eeB

R

QR

πωµσωµ

63

2 0

0 ==

B-field of a rotating charged conducting sphere 19

Conducting sphere (10)

Plot of B for:

Q = 1

µ0 = 1

ω = 1

(in SI-units)

zP / R

zeB 3

20

6

pz

RQ

πωµ=

zeBR

Q

πωµ

6 0=

B-field of a rotating charged conducting sphere 20

Conclusions (1)

Homogeneously charged sphere

(see other presentation)

zeB3

20

10

Pz

RQ

πωµ= ( ) zeB 22

30 35

20 PzR

R

Q −=π

ωµ|zP| < R|zP| > R

Conducting sphere

|zP| > R |zP| < R

zeB3

20

6

pz

RQ

πωµ= zeB

R

Q

πωµ

6 0=

B-field of a rotating charged conducting sphere 21

Conclusions (2)

Plot of B for:

Q = 1

µ0 = 1

ω = 1

(in SI-units)

zP / R

Homogeneously charged sphere

Conducting sphere

The end !!