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1 9/30/15 1 RC Circuits RC Circuits 9/30/15 2 Charging a capacitor: C initially uncharged; connect switch to a at t=0 Calculate current and charge as function of time. Apply Kirchhoff’s Voltage Law: ε q C IR = 0 Short term: Long term: ε 0 I 0 R = 0 I 0 = ε R (q = q 0 = 0) ε q C 0 R = 0 q = Cε (I c = 0) Intermediate term: ε q C dq dt R = 0 9/30/15 3 Solution dq dt = ε R q RC dq ε / R q / RC 0 Q = dt 0 t X = ε / R q / RC dX = 1 RC dq RC dX X ε R ε R Q RC = dt 0 t ln x ε R ε R Q RC = ln ε R Q RC ε R = t RC e t RC = 1 Q ε C 9/30/15 4 Continued τ = RC Q = Cε (1 e t /τ ) Capacitive Time Constant: The greater the , the greater the charging time. V c = Q C = ε (1 e t /τ ) I = dQ dt = ε R e t /τ Units of : F = V A C V = C C/s = s

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Page 1: dX RC dt - physics.purdue.edu

1

9/30/15 1

RC Circuits RC Circuits

9/30/15 2

•  Charging a capacitor:

C initially uncharged; connect switch to a at t=0

Calculate current and charge as function of time.

•  Apply Kirchhoff’s Voltage Law: ε − qC− IR = 0

•  Short term:

•  Long term:

ε − 0 − I0R = 0

I0 =εR

(q = q0 = 0)

ε − q∞C

− 0 ⋅ R = 0 q∞ = Cε

(Ic = 0)

Intermediate term:

ε − qC−dqdtR = 0

9/30/15 3

Solution dqdt

=εR−

qRC

dqε / R − q / RC0

Q

∫ = dt0

t

X = ε / R − q / RC dX =−1RC

dq

−RC dXXε

R

εR−QRC

∫ = dt0

t

∫ ln x εR

εR−QRC = ln

εR− QRC

εR

⎢⎢⎢

⎥⎥⎥=

−tRC

e−

tRC = 1− Q

εC⎛⎝⎜

⎞⎠⎟

9/30/15 4

Continued

τ = RCQ = Cε(1− e− t /τ )

Capacitive Time Constant: "

The greater the , the greater the charging time.

Vc =QC

= ε(1− e− t /τ )

I = dQdt

=εRe− t /τ

Units of :

ΩF =VACV

=CC/s

= s

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9/30/15 5

Charging a Capacitor Q = Cε(1− e− t /τ )

I = εRe− t /τ

t = 0t = ∞t = τ

t = 0t = ∞t = τ

at

at

9/30/15 6

Charging a Capacitor DEMO

9/30/15 7

RC Circuits •  Discharging a capacitor: •  C initially charged with Q=C"•  Connect switch S2 at t=0.

•  Apply Kirchhoff’s Voltage Law: qC+ IR = 0

•  Short term:

•  Long term:

(q = q0 = 0)

(Ic = 0)

Intermediate term: ε + IR = 0

I0 =−εR

q∞ = 0

qC+dqdtR = 0

9/30/15 8

Solution

R dQdt

+QC

= 0 dqq

= −dtRC0

t

∫Cε

Q

−tRC

= lnQ CεQ = ln Q

Q = Cεe−

tRC

I = dQdt

=−εRe−

tRC

Page 3: dX RC dt - physics.purdue.edu

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9/30/15 9

Discharging a Capacitor

DEMO

t = 0t = ∞t = τ

t = 0t = ∞t = τ

at

at

Q = Cεe−

tRC

I = −εRe−

tRC

9/30/15 10

Behavior of Capacitors

•  Charging –  Initially, the capacitor behaves like a wire. –  After a long time, the capacitor behaves like an open

switch in terms of current flow.

•  Discharging –  Initially, the capacitor behaves like a variable battery. –  After a long time, the capacitor behaves like an open

switch

9/30/15 11

Question 3 "

R

R

C

C

"

RRC

C

CIRCUIT 2 CIRCUIT 1

.  1 < 2

.  1 = 2

C. 1 > 2

9/30/15 12

Magnetic Field

•  Large Magnetic fields are used in MRI (Nobel prize for medicine in 2003)

•  Extremely Large magnetic field are found in some stars

•  Earth has a Magnetic Field

Page 4: dX RC dt - physics.purdue.edu

4

9/30/15 13

Bar Magnets

From North to South

N

S

N

S

Attraction

S

N

N

S

Repulsion

•  Bar magnet ... two poles: N and S Like poles repel; Unlike poles

attract. •  Magnetic Field lines: (defined in

same way as electric field lines, direction and density)

NS

DEMO

9/30/15 14

DEMO of Magnetic Field Lines

Magnetic Field Lines of a bar magnet

Electric Field Lines of an Electric Dipole

NS

9/30/15 15

Magnetic Monopoles

•  How can you isolate this magnetic charge?

Try cutting a bar magnet in half:

N S N N S S

•  In fact no attempt yet has been successful in finding magnetic monopoles in nature but scientists are looking for them.

•  One explanation: there exists magnetic charge, just like electric charge. An entity which carried this magnetic charge would be called a magnetic monopole (having + or - magnetic charge).

9/30/15 16

Earth’s Magnetic Field

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9/30/15 17

Earth’s Magnetic Field

Magnetic North Pole 1999

9/30/15 18

Earth’s Magnetic Field

Magnetically Quiet Day

Magnetically DisturbedDay

9/30/15 19

Earth’s Magnetic Field

Since 1904: 750 km, an average of 9.4 km per year.

From 1973 to late 1983: 120 km, an average of 11.6 km per year

Earth’s Magnetic Field