- Please answer all the questions on this test.

- All questions are worth 10 marks each.

- Show all your work.

- For problems involving forces and fields, include free body diagrams.

- Please start each problem on a new page in the exam booklet.

- One 8½ × 11 sheet containing formulas is allowed.

- Formula sheet must be submitted with examination to receive marks.

- Time to do the test is three hours maximum.

- If you have any questions, raise your hand and remain seated.

1. Two capacitors, C₁ = 80 μF and C₂ = 20 μF, are separately, fully charged by a 20 V battery. The two capacitors are then put in a circuit with a resistor, R = 1000 Ω, and an open switch S as shown below.
   (a) What is the charge on each capacitor before being placed in the circuit?
   (b) What is the charge on each capacitor after being placed in the circuit (S is open)? Explain.
   (c) When switch S is closed, how will the total charge stored on the two capacitors, Q_T = Q₁ + Q₂, change with time?
   (d) How is the charge on each capacitor related to Q_T(t)?
   (e) How long with it take for the charge on C₁ to drop to 35% of its original value?
   (f) What will be the charge on C₂ at t = 0.25 seconds after the switch is closed?
   
   

2. A thin ring of radius R has a charge Q uniformly distributed on it.
   (a) From first principles, determine the electric field at a distance along an axis through the centre of the ring and perpendicular to the ring.
   
   
   (b) Derive the potential difference between x = 0 and x = 5R.
   (c) If a charge q = 1.00 μC is released at x =0, how fast will the charge be moving when it gets to x = 5R. The mass of the charge is m = 0.100 kg. Assume Q = 5.00 μC and R = 0.100 m.
   

3. A sphere consists of a hollow centre of radius R and two concentric spherical shells each of thickness R/2. The inner shell carries a uniformly distributed charge of -Q, the outer shell 3Q. What is ρ_inner, the charge density of the inner shell? What is ρ_outer, the charge density of the outer shell? Give expressions for E(r) in all four regions. Sketch E(r) as a function of R.
   
   

4. The diagram below shows a loop of wire carrying a current I. The curved part of the loop is circular with radius R. Distance s is the distance from point P to the horizontal wire segment. The angle α is less than 90°. Using the Law of Biot-Savart, derive an expression for the magnetic field at point P, the centre of the circular arc. Express your answer in terms of μ₀, I, R, s, and α. Hint - consider each piece separately and for the horizontal piece express the limits of integration in terms of s and α.
   

5. The consists of three batteries and three resistors. Resistor R₂ and battery ε₂ are unknown but R₁ = 10 Ω, R₃ = 15 Ω, ε₁ = 12 Volts, and ε₃ = 4 Volts. What would ε₂ have to be so that no current flows through R₂?
   
   

6. The diagram below shows a region of uniform magnetic field B = 0.250 T out of the paper. In the bottom left hand corner of the region is a velocity selector of unknown electric field E which has a small gap to allow only certain particles to pass through. The gap is a distance d = 15.0 cm from a plate. A sample of radioactive cesium is emitting alpha particles ( |q| = 2e and m = 6.6413 × 10^-27 kg) into the velocity selector. What must be the direction and magnitude of the minimum electric field in the velocity selector if the alpha particles are not to hit the plate? Is the ion positive or negative? Explain.
   
   

7. In the diagram below, a flat conducting loop of wire of dimensions h = 30.0 cm by l = 20.0 cm is immersed perpendicularly in a magnetic field which changes with time,
   B(t) = t⁴-3t³-6t²+4t+1 .

   The diagram shows t = 0, where B > 0 indicates that the magnetic field is coming out of the paper. The resistanceless wire is connected to a resistor R = 25.0 Ω. Find
   (a) when the induced emf is a maximum. Determine the magnitude and direction of the current in the loop.
   (b) when the induced emf is a minimum. Determine the magnitude and direction of the current in the loop.
   

   [If you cannot recall how to find extrema, I will tell you for a small deduction of marks.]
   

   

8. The current density in a cylindrical wire of radius R is given by
   
   j(r) = A(1-e^r) ,

   where A is a positive constant. Use Ampere's Law to determine the magnetic field produced by the wire as a function of the radial distance from the centre of the wire. Sketch the result.
   

9. A rectangular loop of wire carries a current of 6.3 A. Nearby in the plane of the loop is a straight wire carrying a current of 5.4 A. What is the magnitude and direction of the net force on the entire loop if the directions and dimensions are as given below in the diagram?
   
   

10. As shown below, two capacitors are connected in parallel with a 9.0-V battery across them. Each capacitor has a plate area of 15 cm by 12 cm and a plate separation of 2.2 mm. One capacitor has an air gap, the other has a nylon dielectric (k = 3.5).
    (a) What is the capacitance of each capacitor?
    (b) What is the charge on each capacitor?
    (c) A switch is opened and the dielectric is then removed from the second capacitor leaving it with an air gap. What are the new charge and potential difference on each capacitor?
    

Formulas

Coulomb's Law and Electric Fields:

F = kq1q2/r2	F= q0E	EPoint Charge = kQ/r2
Ewire = 2kλ/r	Eplate = Σ/2ε0	Esphere = KQ/r2

Gauss' Law

	φE = òE ·ndA	= Qinside/ε0
cylinders -	E(a)2πaL	= [2πL òr(r)rdr]/ε0
spheres -	E(a)4πa2	= [4pòr(r)r2dr]/ε0

Electrostatic Potential

ΔV = V(b) - V(a) = -òbaE·dl = -òbaE(r)dr
Vpoint charge = kQ/r	V = V1 + V2 + ...	ΔK =ΔU = -qΔV

Capacitors

C = Q/V	CP = C1+C2	1/Cs = 1/C1+1/C2
Cplates = ε0A/d	Eplates = Q/ε0A	Cdielectric = kC0
Edielectric = kE0	Σb = [(k-1)/k]sf	U = ½Q2/C = ½QV = ½CV2
charging:	Q = Cε(1 - e-t/RC)	I = (ε /R)e-t/RC
discharging:	Q = Q0e-t/RC	I = (ε /R)e-t/RC

Resistors

Rs = R1 + R2	1/Rp = 1/R1 + 1/R2
V = IR	P = I2R = V2/R = VI
åEi -åIiRi = 0	åqin =åqout

Magnetic Field

F = qv×B	F = IL×B	r = mv/qB
f = 1/T = qB/2πm	vselector = E/B	m = NIAn
τ = m×B	B = (μ0/4πr2)qv×r
dBBiot-Savart = (μ0/4πr2)IdL×r		Bwire = μ0I/2πR
Bloop = μ0I/2R	Bsolenoid = μ0nI

Ampere's Law

òCB·dl		= μ0Iinside
cylindrical wires:	B2πa	= 2pm0òj(r)rdr

Magnetic Induction

φm = BAcosθ

Einduced = dφm/dt

|Emotional| = vBl

Constants

k = 8.99 × 109 N m2/C2

ε0 = 8.85 × 1012 F/m

μ0 = 4π × 107 T-m/A

Identities and Integrals

if ax2+bx+c=0, x = {b±[b2- 4ac]½}/2a	S = Rθ, dS = Rdθ
òxndx = xn+1/(n+1) + C	òx-1dx = lnx + C
òebxdx = ebx/b + C	òxebxdx = (x - b)ebx + C
òcosΩxdx = (sinΩx)/Ω + C	òsinΩxdx = -(cosΩx)/Ω + C
ò(a/[a2+x2]3/2)dx = 1/[a[a2+x2]1/2] + C	ò(x/[a2+x2]3/2)dx = -1/[a2+x2]1/2 + C

Questions? mike.coombes@kwantlen.ca