Electromagnetism
Physics 15b
Lecture #10
RC Circuits
Magnetic Field
Purcell 4.11, 5.1–5.2, 6.1–6.2
What We Did Last Time
Power dissipation in resitors P = IV = I R
Electromotive forces (emfs)
2

Batteries are made of an emf and an internal resistance
Resistor arithmetic Rseries = ∑ R1
i
1
Rparallel
=∑
i
1
Ri
Kirchhoff’s rules
∑I
in

= ∑ Iout
Junction Rule
∑V = 0
Loop rule requires attention to the polarity
Tévenin and Norton equivalences
Open-circuit voltage and short-circuit current
  Req can be evaluated by emf = 0Ω

Loop Rule
loop
VTh = VO
INo = IS
RTh = RNo = VO IS
1
Today’s Goals
Discuss capacitors in DC circuits

Analyze a simple RC circuit
Introduce magnetic field B
Defined by Lorentz force on moving
charges
  Cyclotron motion

Electric current produces B field

Force between two currents
Mathematical properties of B
curl B and div B
  Remaining two of Maxwell’s
equations
  Uniqueness theorem

Hans Ørsted (1777-1851)
Capacitors in DC Circuits
Two plates of a capacitor are insulated from each other

Current cannot flow through it
In a DC circuit, where the current is constant, capacitors
don’t do anything – as if it’s not there
The story is different if the current is not constant
Alternate Current (AC) circuits often use capacitors
  When a switch is thrown (opened or closed) in a DC circuit, the
current changes momentarily before settling into a new constant
value
  Capacitors do something in the process

Let’s look into this transient process
2
Simple RC Circuit
Definition of capacitance : Q = CV
With the switch open, the current I must be zero

Assume Q = 0 as well
−Q
Close the switch at time t = 0
Initially, C has zero charge, so the
potential difference around C is 0
  Kirchhoff’s loop rule says
C
+Q

E + 0 − IR = 0

E!
−
E
R
The current brings electric charge into the capacitor
dQ
=I
dt

I(t = 0) =
+
I
R
Capacitor C is being charged up
by the current I
As Q increases, potential difference V = Q/C appears around C
Simple RC Circuit
V = Q/C
Time t after closing the switch
C holds Q = Q(t)
  Kirchhoff’s loop rule
−Q

E−

Q(t)
− I(t)R = 0
C
+Q
+
I
R
E!
−
Combine with
dQ(t)
= I(t)
dt

C
dQ(t) Q(t)
+
−E =0
dt
RC
Solution to this differential equation is
Q(t) = −ke
−
t
RC
+ EC
Q(0) = 0
(
)
Q(t) = EC −e −t RC + 1
I(t) =
dQ(t) E −t RC
= e
dt
R
3
Simple RC Circuit
(
)
Q(t) = EC −e −t RC + 1
I(t) =
E −t RC
e
R
Solution is an exponential decay with
time constant RC
−Q
C
+Q
I
R
+
E!
−
E
R
EC
t
RC
t
RC
Product RC has the dimension of time (really?)
  RC circuits “relax” exponentially toward the stable equilibrium

You can (sometimes) guess the solution by knowing the
initial condition and the asymptote
Magnets
We all know: magnets have N and S poles
Opposite poles attract, same poles repel, each other
  N points north, S points south
  That means Earth’s North Pole is S, and its South Pole is N
  Is that confusing?

Similar to electric charges N  +, S  −
Difference: magnetic N-S poles cannot be separated


A single magnetic charge (= monopole) does not exist
Cutting a magnet in the middle results in two magnets, each with NS poles
N
S
N
S
N
S
4
Magnetic Field
Just like electric field, we can define magnetic field from the
force on the N pole of a test magnet
S
Magnetic field lines run from
N poles to S poles
  This is cumbersome because
we don’t have a monopole as
a test “charge”

N
N
S
A better definition of magnetic
field uses force on a charged
particle in motion

That’s the Lorentz Force
Lorentz Force
A particle with charge q moving with a velocity v in a
magnetic field B receives a force
B
q
F = v ×B
c
v
F
This is an empirical “fact”, which we use to define B
  Factor 1/c will become natural later

If both E and B fields are present

Note E and B have the same unit in CGS
[electric field] = [magnetic field] =

F = qE +
SI unit for B is tesla [T]
  1 tesla = 104 gauss
q
v ×B
c
dyne statvolt
=
= gauss (G)
esu
cm
F = qE + qv × B
no 1/c in SI
5
Cyclotron Motion
A charged particle is flying in a uniform B field

Assume v and B are perpendicular and B points out of the screen
q
F = v × B = ma
c

F
q
B
v
Constant acceleration perpendicular to v
Particle flies in a circle
m

v 2 qvB
=
r
c
r=
mvc
qB
This is called the cyclotron motion
How long does it take to make a full circle?
v qB
2π r 2π mc
the cyclotron frequency
=
or ω =
T=
=
r mc
v
qB
Magnetic Bottle
Charged particles travel in helix wrapped around magnetic
field lines

Circular motion perpendicular to B
plus linear motion parallel to B
It is possible to confine the path by B field that’s weak in
the middle and strong at the ends
Magnetic bottle is used in
nuclear fusion research
for confining plasma
  There is also a big one
out there in the sky

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Van Allen Belt
Earth’s magnetic field bends and traps charged particles
from the space (= cosmic rays)

We’d die from radiation without it
Trapped particles form the
Van Allen belt

Discovered by Van Allen
using a Geiger counter
aboard Explorer 1 satellite
Edge of the belt closer to
Earth near the poles

Auroras
ISS Expedition 6, NASA
7
Ørsted’s Discovery
Current flowing in a wire makes the needle of a nearby
compass swing
I
Electricity and magnetism are related


B field points around the wire
Strength of B decreases as 1/(distance)
B
r
I
B = (const) ×
r
How can we determine the constant?


I
Bring a moving charge near the wire
Easier solution: bring another wire and
run current on it
Force Between Currents
Run two straight wires in parallel
  I1
I1
I2
creates rotating B field
B=k
I1
r
carrier charge
F
q
  I2 feels the force F = nL
v ×B
c
# of carriers per unit length
L
carrier velocity
I
II L
q
F = nL v ⋅ k 1 = k 1 2
c
r
cr
r
Direction: Parallel currents attract each other

If I1 and I2 flow in opposite directions, they repel each other
Measurement can tell us the value of k
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Constants and Units
The force turns out to be F =

2I1I2L
i.e. k = 2/c (in CGS)
  We’ll see why in Lecture 13
I1
I2
2
c r
F
2I
Magnetic field due to current I is B =
cr
  Including the direction
B=
2I × r̂
cr
Vector I points the
direction of the current
In SI, we find B =
F=
L
B
r
µ0I
permeability
µ0 = 4π × 10 −7 T ⋅ m A of vacuum
2π r
µ0I1I2L
For I1 = I2 = 1 A, r = L = 1 m, F = 2×10–7 N exactly
2π r
Because ampere (A) is defined that way
Divergence and Curl
Current I is coming out of the screen
2I × r̂ 2I
B=
= φ̂
cr
cr

Except at r = 0, B satisfies
div B = 0

curlB = 0
B
r
φ
I
Recall
∇ ⋅F =
1 ∂(rFr ) 1 ∂Fφ ∂Fz
+
+
r ∂r
r ∂φ
∂z
⎛ 1 ∂F ∂Fφ ⎞
⎛ ∂(rFφ ) ∂F ⎞
⎛ ∂Fr ∂Fz ⎞
z
∇×F = ⎜
−
−
φ̂ + ⎜
− r ⎟ ẑ
⎟ r̂ + ⎜
⎟
r
∂
φ
∂z
∂z
∂r
∂r
∂φ ⎠
⎝
⎠
⎝
⎠
⎝
What happens at r = 0?
9
Div and Curl at r = 0
Apply divergence theorem to a cylinder around I

B is parallel to the surfaces
∫
V

B
∇ ⋅B dv = ∫ B ⋅ da = 0
S
We know ∇⋅B = 0 at r > 0  ∇⋅B = 0 at r = 0 as well
Apply Stokes’ theorem to a circle around I

B is parallel to the loop
∫
S
(∇ × B) ⋅ da = ∫ B ⋅ d s =
C
2I
4π I
ds =
C cr
c
∫
We know ∇×B = 0 at r > 0  ∇×B = ∞ at r = 0
  Take into account finite radius of the wire

I=
∫
wire
J ⋅ da
implies
∇×B =
4π J
inside the wire
c
Generalize
B field generated by any current density J(x,y,z) satisfies
div B = 0

curlB =
4π J
c
curlB = µ0 J in SI
Compare with E field due to charge density ρ(x,y,z)
div E = 4πρ
curlE = 0
These are Maxwell’s equations for static fields
For a given distribution of J (or ρ), these equations uniquely
determine the B (or E) field

This is another uniqueness theorem — Let’s prove it
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Proof of Uniqueness
Suppose B1 and B2 satisfied ∇⋅B = 0 and ∇×B = 4πJ/c



Let D ≡ B1 − B2, then D satisfies ∇⋅D = 0 and ∇×D = 0
Since curl is zero, D can be expressed as a gradient of a scalar
field, i.e., D = ∇f(x,y,z)
Take a div. and we find ∇×D = ∇2f(x,y,z) = 0
  f(x,y,z) satisfies Laplace’s equation
Inject physics: the current density J has a finite extent
At large distance, B1 and B2 approach zero, and so does D
  f(x,y,z) approaches a constant at large distance
  Since f(x,y,z) satisfies Laplace, it cannot have a min or max
  f(x,y,z) = const. everywhere
  D = 0 everywhere  B1 = B2

Caveat: this may not work for unrealistic (e.g. infinite) J
Summary
RC circuits relax exponentially toward a steady state

According to exp(−t/RC), where RC is the time constant
Lorentz force on a moving charge F = qE +

Defines magnetic field B
Infinite line current I creates B =

Two such currents
2I I l
attract each other by F = 12 2
2I × r̂
cr
q
v ×B
c
I1
I2
F
c r
Maxwell’s equations for static current/charge
4π J
div B = 0 curlB =
div E = 4πρ curlE = 0
c

B
l
r
“Uniqueness” of solutions B and E
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