Name: ________________________________________________________ 1. Calculating Potential by Direct Integration (10 points). The wire
in the figure has linear charge density λ. What is the electric potential at
the center of the semicircle? SHOW YOUR WORK. Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 2. Capacitance and Energy (10 points). A capacitor has square
metal plates that are 3.0 cm on a side, with a separation of about 5.0
mm. The potential difference between the plates is 25.0 V. The plates
are close enough that we can ignore fringing at the ends. Under these
conditions: (a) how much charge is on each plate, and
(b) how strong is the electric field between the plates?
(c) what is the electric potential at 3.0 mm from the negative plate?
(d) If an electron is ejected at rest from the negative plate, how fast is it
moving when it has travelled 3.0 mm?
(e) how fast is it moving when it reaches the positive plate?
(f) what is the energy stored in this capacitor (at 25V)? Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 3. DC Circuits (10 points). For the circuit, find the current through and
the potential difference V across each resistor. Place your results in a
table for ease of reading, and show – in a sequence of organized and
clear diagrams – how you are calculating equivalent resistances
throughout this circuit. Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 4. RC Circuits (10 points). The switch in the figure has been closed for
a very long time.
a. What is the charge on the capacitor?
b. The switch is opened at t = 0 s. At what time has the charge on the
capacitor decreased to 10% of its initial value?
c. What is the current though the 10 resistor at the time you found in
part (b)? Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 5. DC Circuits (10 points). What is the electric potential at the points A,
B, and C in the circuit below? V1 = 20.0 volts. V2 = 10.0 volts. V3 = 5.00
volts. R1 = 1.00 Ω. R2 = 2.00 Ω. R3 = 3.00 Ω. R4 = 4.00 Ω.
Note that the circuit is grounded where indicated, and SHOW YOUR
WORK! Physics 200, E&M, Spring 2017 Name: ________________________________________________________ ⃗⃗ (10 points). A Geiger-Mueller tube is a radiation
6. Getting V from detector that consists of a very long, hollow metal cylinder (the cathode)
of inner radius ra and a concentric thin metal rod of radius rb (the
anode), as shown in the figure. The charge per unit length on the anode
is and the cathode is -, and a gas fills the space between the
electrodes. When a high-energy elementary particle passes through this
space, it can ionize a molecule of the gas, and the strong electric field
makes the resulting ion and electron accelerate in opposite directions.
They strike and ionize other molecules of gas, causing an avalanche of
electrical discharge which is counted by an external circuit.
(a) Show that (in the middle of the
cylinder far away from either end)
the magnitude of the electric
potential difference between the
wire and the cylinder, V, is: |∆| = 2 ( ) (b) The absolute value sign on V is because the negative sign in the
definition of V, etc., can be confusing (e.g., you must define the positive
direction correctly for the integral, etc.). But, given the direction of the E
field and the charge distribution, is the inner cylinder at a higher or
lower voltage than the outer cylinder? EXPLAIN YOUR ANSWER IN
BULLETPOINT FORM. Physics 200, E&M, Spring 2017 Name: ________________________________________________________ ⃗⃗ from V
7. Electric Potential by Direct Integration & Getting (10 points). As shown in the figure below, a disc of radius R has a
nonuniform surface charge density of = Cr, where C is a constant and r
is the distance from the center of the disc to a spot on the disc.
(a) What is the total charge on the disc Qdisc? SHOW YOUR WORK!
(b) Find, by direct integration, the electric potential at an observation
point P located a distance x from the disc along its axis.
(c) Find the electric field E along the axis of the disc (Hint: E = − ̂). Your expression will be complicated, but demonstrate to me that it has
the proper units for electric field.
-- OR -(d) Show that your expression for V from part (b) can be made to look
like the V of a point charge with the Qdisc when x/R >>> 1. In taking this
limit, you will have to “massage” your expression from part (b) in just
right ways…show those steps and justify them. Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 8. Capacitance and Energy (10 points). The circuit in the figure
consists of two identical parallel metal plates connected to two identical
metal springs , a switch, and a 100V battery. With the switch open, the
plates are uncharged and separated by a distance d = 8.00 mm, and have
a capacitance C = 2.00 f. When the switch is closed, the distance
between the plates decreases by a factor of 0.5.
(a) How much charge collects on each of plate?
(b) What is the spring constant k for each spring?
(c) What is the force that one plate exerts on the other? SHOW YOUR
WORK!! Physics 200, E&M, Spring 2017 Name: ________________________________________________________ 9. BONUS PROBLEM (20 points). NOT DUE UNTIL 04/19 !!!
Reconsider the capacitor and spring circuit/system of Problem #8. Let
the capacitor fully charge as in problem 8(a) so that the springs are
stretched and at equilibrium as in 8(a) and 8(b).
Now open the switch so that no current can flow, and replace the 100 V
battery with a resistor R = 1 M (1,000,000 ), and then close the
switch at t = 0. As the capacitor discharges, the plates of the capacitor
will “relax” back towards their initial separation of 8.00 mm.
Derive an expression for (i) the charge on the capacitor as a function of
time and (ii) the separation between the plates as a function of time.
Work symbolically (i.e., nevermind the numerical values given), and you
will have to solve a differential equation.
This is not a simple RC circuit problem…the capacitance is changing as a
function of time because the plate separation is changing. This is a hard
problem, and it might not even have an analytical solution. In that case –
to get full credit – numerically/graphically solve it using the values
listed. Physics 200, E&M, Spring 2017

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