Electricity and Magnetism
Review 2: Units 7-11
Mechanics Review 2 , Slide 1
Example: RC Circuit
R1
V
S
R2
R3
C
In this circuit, assume V, C, and Ri are
known. C is initially uncharged and then
switch S is closed.
What is the voltage across the capacitor
after a very long time ?
Immediately after S is closed:
what is VC, the voltage across C?
what is I2, the current through R2?
At t = 0 the capacitor
behaves like a wire. Solve
using Kirchhoff’s Rules.
R1
V
VC = 0
S
R2
R3
Example: RC Circuit
I1
S
R1
R2
R3
In this circuit, assume V, C, and Ri are
known. C is initially uncharged and then
switch S is closed.
C
V
What is the voltage across the capacitor
after a very long time ?
Immediately after S is closed, what is I1, the current through R1 ?
S
R1
R2
R3
V
VC = 0
I1 =
V
RR
R1 + 2 3
R2 + R3
Example: RC Circuit
S
R1
In this circuit, assume V, C, and Ri are
R3 known. C is initially uncharged and then
switch S is closed.
R2
C
V
What is the voltage across the capacitor
after a very long time ?
After S has been closed “for a long time”, what is I2, the current
towards C ?
I
R1
V
I2 = 0
VC
R3
I2 = 0
After a long time the capacitor
and R2 are are not connected to
the circuit.
Example: RC Circuit
S
R1
In this circuit, assume V, C, and Ri are
R3 known. C is initially uncharged and then
switch S is closed.
R2
C
V
What is the voltage across the capacitor
after a very long time ?
I
I
R1
VC
V
R3
R3
V
R1 + R3
VC = V3 = IR3 = (V/(R1 + R3))R3
Example RC Circuit
S
R1
V
R2
R3
C
In this circuit, assume V, C, and Ri are
known. C initially uncharged and then
switch S is closed for a very long time
charging the capacitor. Then the switch is
opened at t = 0.
What is tdisc, the discharging time constant?
What is the current on R3 as a function of time?
Redraw the circuit with the switch open. Now it looks like a
simple RC circuit.
R2
t c = RC = ( R2 + R3 ) C
C
R3
q(t) = q(0)e-t/RC q(0) = CVc
dq
I=
dt
Example: Capacitors
Three capacitors are connected to a battery as shown.
A) What is the equivalent capacitance?
B) What is the total charge stored in the system?
C) Find the charges on each capacitor.
Parallel:
C23 = C2 + C3
Series:
(1/C123) = (1/C23) + (1/C1)
Total Charge:
Q = C123 V
Charges on capacitors:
Q1 = Q
Q2 + Q3 = Q
V2 = V3
V
Example: Kirchhoff’s Rules
R1
+
+
V1
+ V2
R2
I2
- + R3
-
I1
V3
+ + -
In this circuit Vi and Ri are known.
What are the currents I1 , I2 , I3?
I3
Label and pick directions for each current
Label the + and - side of each element
Batteries are easy
For resistors, the “upstream” side is +
Example: Kirchhoff’s Rules
R1
-
R2
V1
I1
+ + V2
I2
- + -
+
R3
V3
In this circuit Vi and Ri are known.
What are the currents I1 , I2 , I3?
I3
- ++ Kirchhoff’s Rules give us the following 4 equations:
1. I2 = I1 + I3
2. - V1 + I1R1 - I3R3 + V3 = 0
3. - V3 + I3R3 + I2R2 + V2 = 0
4. - V2 - I2R2 - I1R1 + V1 = 0
We need 3 equations:
Which 3 should we use?
The node equation (1.) and any
two loops.
Example: Calculating Capacitance
E field produced
AFirst
soliddetermine
cylindrical conductor
of radiusbya charged conductors:
length l and charge Q is coaxial with a thin
cylindrical shell of radius b and charge –Q.
Assume l is much larger than b.
Find the capacitance
Integrate E to find the potential difference V
E(r) =
2kl 2kQ
=
r
rl
V = -2k
Q b
ln( )
l
a
Q
ℓ
C= =
V 2ke ln(b / a)
Example: Calculating Capacitance
A spherical
conducting
of
First
determine
E fieldshell
produced
by charged conductors:
radius b and charge –Q is
concentric with a smaller
conducting sphere of radius a
and charge Q.
Find the capacitance.
Integrate E to find the potential difference V
E(r) =
kQ
r2
1 1
V = kQ( - )
b a
Q
ab
C= =
V ke ( b - a)
Example: Capacitors
In the circuit shown the switch SA is originally closed and the
switch SB is open.
(a) What is the initial charge on each capacitor.
Then SA is opened and SB is closed.
(b) What is the final charge on each capacitor.
(c) Now SA is closed also. How much additional charge flows
though SA?
SA
SB
Initial Charge:
Q1i = C1 ΔV Q2i = Q3i = 0
Final Charge:
Q1f = C1 ΔVf
Q2f = Q3f
ΔVf = Q2f /C2 + Q3f /C3
C2
ΔV
C1
Q1i = Q1f + Q2f
Both switches closed:
Qtotal = Ctotal ΔV
Ctotal = C1+1/(1/C2+1/C3)
C3
Example: Capacitor with Dielectric
An air-gap capacitor, having capacitance C0 and width x0 is
connected to a battery of voltage V. A dielectric (k ) of width
x0/4 is inserted into the gap as shown. What is Qf, the final
charge on the capacitor?
V
x0
C0
V
k
C
x0/4
What changes when the dielectric added? C and Q change,
V stays the same.
Q
V
Example: Capacitor with Dielectric
Can consider capacitor to be two capacitances, C1 and C2, in parallel
k
C1
=
What is C1 ?
C1 = 3/4C0
What is C2 ?
C2 = 1/4 k C0
What is C ?
C = C0 (3/4 + 1/4 k)
What is Q?
Qf = VC0 (3/4 + 1/4 k)
k
C2
Q
C
V
For parallel plate capacitor: C = e0A/d
Example: Circuits
R2
R1
V
R3
R4
In the circuit shown: V = 18V,
R1 = 1W, R2 = 2W, R3 = 3W, and R4 = 4W.
What is V2, the voltage across R2?
First combine resistors to find
the total current:
R2 and R4 are connected in series (R24) which is connected in
parallel with R3
R1
Redraw the circuit using
the equivalent resistor R24
= series combination of R2
and R4.
V
R3
R24 = R2 + R4 = 2W + 4W = 6W
R24
Example: Circuits (Without Kirchhoff’s Rules)
R1
V
R3
In the circuit shown: V = 18V,
R1 = 1W, R2 = 2W, R3 = 3W, and R4 = 4W.
What is V2, the voltage across R2?
R24
R3 and R24 are connected in parallel = R234 = 2 W
1/Req = 1/Ra + 1/Rb
1/R234 = (1/3) + (1/6) = (3/6) W -1
R1 and R234 are in series. R1234 = 1 + 2 = 3 W
R1
Ohm’s Law
V
V
R234
= I1234
R1234
I1 = I1234 = V/R1234
= 6 Amps
Example: Circuits
V
= I1234
R1
V
I1 = I234
R1234
a
In the circuit shown: V = 18V,
R1 = 1W, R2 = 2W, R3 = 3W, and R4 = 4W.
R24 = 6W
R234 = 2W I1234 = 6 A
What is V2, the voltage across R2?
Since R1 in series with R234
I234
RR234
234
b
= I1234 = I1 = 6 Amps
V234
What is V234 (Vab)?
= I234 R234 = 6 x 2 = 12 Volts
Example: Circuits
R1
I1234
V
R1
I24
R2
V
R3
R3
R24
R4
I24 = V234 / R24 = 2Amps
I24 = I2 = 2Amps
Ohm’s Law
V2 = I2 R2
= 4 Volts
Example: Circuits
I1
I2
R1
R2
R1
I3
V
=
R3
a
V
R234
R4
b
What is I3 ?
I1 = I2 + I3
I3 = 4 A
What is P2 ?
P2 = I2V2 = I22 R2 = 8 W
V = 18V
R1 = 1W
R2 = 2W
R3 = 3W
R4 = 4W
R24 = 6W
R234 = 2W
V234= 12V
V2 = 4V
I1 = 6 Amps
I2 = 2 Amps
Example: Kirchhoff’s Rules
Given the circuit below. Use Kirchhoff’s rules to find the
currents I1, I2 and I3, and the charge Q on the capacitor.
What is the voltage difference between points g and d?
(Assume that the circuit has reached steady state currents)
I1 = 1.38 Amps
I2 = - 0.364 Amps
I3 = 1.02 Amps
Q = 66.0 μC
Vgd = 2.90 V