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2011 Electricity and Magnetism II
E&M. 2.
The circuit represented above contains a 9.0 V battery, a 25 mF capacitor, a 5.0 H inductor, a 500 W
resistor,
and a switch with two positions, S1 and S2 . Initially the capacitor is uncharged and the switch is open.
(a) In experiment 1 the switch is closed to position S1 at time t1 and left there for a long time.
i. Calculate the value of the charge on the bottom plate of the capacitor a long time after the switch
is closed.
ii. On the axes below, sketch a graph of the magnitude of the charge on the bottom plate of the capacitor
as a function of time. On the axes, explicitly label any intercepts, asymptotes, maxima, or minima with
numerical values or algebraic expressions, as appropriate.
iii. On the axes below, sketch a graph of the current through the resistor as a function of time. On the
axes,
explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic
expressions, as appropriate.
(b) In experiment 2 the capacitor is again uncharged when the switch is closed to position S1 at time t1 .
The switch
is then moved to position S2 at time t2 when the magnitude of the charge on the capacitor plate is 105 mC,
allowing electromagnetic oscillations in the LC circuit.
i. Calculate the energy stored in the capacitor at time t2 .
ii. Calculate the maximum current that will be present during the oscillations.
iii. Calculate the time rate of change of the current when the charge on the capacitor plate is 50 mC.
2008 Electricity and Magnetism II
In the circuit shown above. A and B are terminals to which different circuit components can be connected.
(a) Calculate the
potential difference
across R 2, immediately
after the switch S is
closed in each of the
following cases.
i.
A 50 Ω resistor
connects A and B.
ii.
A 40 mH inductor
connects A and B.
iii. An initially
uncharged 0.80 pF
capacitor connects A and
B.
(b) The switch gets closed at time t = 0. On the axes below. sketch the graphs of the current in the
100 Ω resistor R3, versus time t for the three cases. Label the graphs R for the resistor, L for the inductor,
and C for the capacitor.
2000- Electricity and Magnetism I
Lightbulbs A, B. and C are connected in the circuit shown
above.
a. List the bulbs in order of their brightness, from
brightest to least bright. If any bulbs have the same
brightness. state which ones. Justify your answer.
Now a switch S and a 5.0 mH inductor are added to the circuit
as shown above. The switch is closed at time t = 0.
b. Determine the currents I A,I B and I C , for the following
times.
i. Immediately after the switch is closed.
ii. A long time after the switch is closed.
a. On the axes below, sketch the magnitude of the potential difference
VL across the inductor as a
function of time, from immediately after the switch is closed until a long time after the switch is closed.
b. Now consider a similar circuit with an uncharged
5.0µF capacitor instead of the inductor, as shown
above. The switch is again closed at time t = 0. On the axes below, sketch the magnitude of the
potential difference Vcap across the capacitor as a function of time, from immediately after the
switch is closed until a long time after the switch is closed.
1996- Electricity and Magnetism II
Capacitors 1 and 2, of capacitance C=4μF and
C2= 12μF, respectively, are connected in a
circuit as shown above with a resistor of
resistance R=1000Ω and two switches.
Capacitor 1 is initially charged to a voltage Vo
= 50 V, and capacitor 2 is initially uncharged. Both of the switches S are then closed at time t = 0.
a. What are the final charges on the positive plate of each of the capacitors 1 and 2 after equilibrium
has been reached?
b. Determine the difference between the initial and the final stored energy of the system after
equilibrium has been reached.
c. Write, but do not solve, an equation that, at any time after the switches are closed, relates the charge
on capacitor Cl, its time derivative (which is the instantaneous current in the circuit), and the
parameters Vo, R, C1, and C2.
The current in the resistor is given as a function of time by I = Ioe -1/τ, where Io= 0.5A and τ= 3 x 10-4s.
d. Determine the rate of energy dissipation in the resistor as an explicit function of time.
e. How much energy is dissipated in the resistor from the instant the switch is closed to when
equilibrium is reached?
1986- Electricity and Magnetism II
Five resistors are connected as shown above to a 25volt source of emf with zero internal resistance.
a. Determine the current in the resistor labeled R.
A 10-microfarad capacitor is connected between
points A and B. The currents in the circuit and the
charge on the capacitor soon reach constant values.
Determine the constant value for each of the
following.
b. The current in the resistor R
c. The charge on the capacitor
The capacitor is now replaced by a 2.0-henry inductor with zero resistance. The currents in the circuit
again reach constant values. Determine the constant value for each of the following.
d. The current in the resistor R
e. The current in the inductor
Homework
A section of a long conducting cylinder with inner radius a and outer radius b carries a current I0 that has
a uniform current density, as shown in the figure above.
(a) Using Ampère’s law, derive an expression for the magnitude of the magnetic field in the following
regions as a function of the distance r from the central axis.
i. r < a
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ii. a < r < b
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iii. r = 2b
(b) On the cross-sectional view in the diagram above, indicate the direction of the field at point P, which
is at a distance r = 2b from the axis of the cylinder.
(c) An electron is at rest at point P. Describe any electromagnetic forces acting on the electron. Justify
your answer.
Now consider a long, solid conducting cylinder of radius b carrying a current I0 . The magnitude of the
magnetic field inside this cylinder as a function of r is given by B = 0I0r/ 2b2 . An experiment is
conducted using a particular solid cylinder of radius 0.010 m carrying a current of 25 A. The magnetic
field inside the cylinder is measured as a function of r, and the data is tabulated below.
ii. Use the slope of your line to estimate a value of the permeability .