Math. Methods of Physics – Winter Final
PhySys/0607/quizzes/SpfFin
This is a CLOSED-BOOK, take-home exam.
This is designed as a two-hour exam. You have 3 hours.
Pace yourself, and take breaks when you need them.
SHOW ALL YOUR WORK, explain your reasoning, and include units when necessary, to
receive full credit
Please circle or underline your answers when appropriate, for clarity.
Keep answers in simplest exact form or make order-of-magnitude estimates.
(sign legibly)_______________________________________________________________
I affirm that I have worked this exam with WITHOUT using a calculator, text, HW,
quizzes, computer, classmates, or other resources.
Possibly useless information (please ask if you need more info)
G = 6.67x10-11 N m2/kg2
0 = 8.85 x 10-12 C2/N.m2
g=9.8 m/s2
c=3x108 m/s h = 6.63 x 10-34 J.s
 = 5.67 x 10-8 W/m2.K4
k = 1.38 x10-23 J/K
F = mg
F = -GmM/r2 F = -qQ/(40r2)
F = -kx
F = dp/dt = ma
p = mv
s = r
v = r
= ½ I2
K= ½ mv2
I = dq/dt
W=qV
E = -dV/dx
F = qvxB = ILxB
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Part I – Calculus
Part II – Research
Part III – Universe
Part IV – Electric and Magnetic fields
Part V – Electromagnetism
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total
F = -dU/dx
a = r
Part IV - Electrostatics and magnetism: (30-45 minutes)
1. What causes electric fields? Give at least two different phenomena with their equations.
2. What causes magnetic fields?
Give at least two different phenomena, and one equation.
3. How do electric fields affect charges? Give a specific illustration with an equation.
4. How do magnetic fields affect charges? Give a specific illustration with an equation.
5. How can you tell if a charge’s motion is caused by an electric or magnetic force?
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6. Check all the fields that could possibly be present, for each charge motion.
E alone
B alone
E and B together
a. Charge at rest
b. Charge moving at a constant speed
c. Charge accelerating in a straight line
d. Charge bending at constant speed
e. Charge bending and accelerating
7. Describe how potentials V are related to electric fields E,
(a) qualitatively and
(b) quantitatively
8. Is an electric field stronger where equipotentials are concentrated or spread out? Explain.
9. Is an electric field stronger near a pointy object or a smooth one? Explain.
10. Is it safer to stand up or crouch down in a lightening storm? Explain.
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Part V: ELECTROMAGNETISM (30-45 minutes)
1. Draw the direction of the magnetic force for each situation below. Assume positive charges.
PhySys\0607\quizzes\SprMidQM
2. A device measures the mass of charged particles by first determining their velocity and
then deflecting them with a magnetic field.
(a) Velocity selector: When a charged particle travels through crossed E and B fields (both of
which fill the region below), the trajectory will be undeflected (continuing in a straight line)
only if the velocity has a certain relationship to E and B. This setup effectively “selects”
particles with velocity v.
uniform B
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
uniform
E
(a) Derive the relationship between E, v, and B
(b) that will select undeflected particles of velocity v.
2 (b) If the electric field E doubles, how will the selected particle’s ENERGY change? It
will: (show your work)
Halve - stay the same – double – quadruple - other
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Now that we know the speed v of the particle from the E and B settings, we’d like to determine
the mass m of the particle – which is the point of this device.
Magnetic deflection:
When a charged particle of velocity v enters a region of perpendicular magnetic field (B)
(where E=0 now), how is it deflected?
(c) Sketch the path of a positively charged particle.
(d) Derive an expression for its radius of curvature in terms of q, v, and B.
(e) Find the mass of the particle in terms of measurable quantities (r, E, and B) and the
charge q.
uniform B
v
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
(f) If the magnetic field B in the deflection region doubles, how will the RADIUS of
curvature of the particle’s path change? It will:
(show your work)
Halve - stay the same – double – quadruple - other
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PhySys/0607/quizzes/WinMidEMMod
1. (a) Find charge distribution q(r) inside a sphere which carries a charge density
proportional to the distance from the origin,  = c r, for some constant c. [Hints: This charge
density is not uniform, and you must integrate to get the enclosed charge q(r<R). A spherical
volume element is d= r2 sin dr d d where (0<<) and (0<).]
(b) Find the electric field inside the sphere.
(c) What is the total charge Q in the sphere? Express the electric field outside the sphere in
terms of Q.
(d) How could you find the energy in this charge configuration? You need not calculate it, but
set it up.
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PhySys/0607/quizzes/WinMidVector
1. Is each function below a vector or a scalar?
(a) v = x sin y xˆ + cos y yˆ + x y zˆ
(b) T  e5 x sin 4 y cos3z
2. What is the definition of the function del or  (in Cartesian coordinates)?
3. (a) Can one find the Laplacian (  2 ) of a scalar or vector function?
(b) Find the Laplacian of the appropriate function from (1) above.
4. (a) Can one find the divergence and curl of a scalar or vector function?
(b, c) Find the divergence and curl of the appropriate function from (1) above.
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(Griffiths 2.35 p,101)
A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell
with inner radius a and outer radius b, as drawn below. The outer shell carries no net charge.
(a) Find the surface charge density  at R, at a, and at b.
(b) Find the electric field E everywhere.
(c) If the outer shell is touched to a grounding wire, how do your answers change? Why?
(d) Find the capacitance of the system.
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Boas 8.3.17 (p.404)
Find the general solution of RI 
(Hint:
I  dq
dt
q
 V for an RC circuit with V  V0 cos t .
C
. Don’t confuse current with your integrating factor.)
.
Boas 8.4.18 : Solve y ' 
y x
.
2 yx
2
2
(Hint: Bernoulli eqn: Let z  y n 1 …)
Arfken p.947: Lagrange multipliers: Consider a particle of mass m in a 3D quantum mechanical
well. The well is a rectangular box with sides a, b, and c. The ground state energy of the particle
h2  1 1 1 
is given by E 
   .
8m  a 2 b2 c 2 
Find the shape of the box (a, b, and c) that will minimize the energy E, subject to the constraint
that the volume is constant.
Boas 7.5.7 (p.355), 7.7.7 (p.360)
0,    x  0
(a) Sketch the PERIODIC function f ( x)  
.
 x, 0  x  
(b) Is it even, odd, or neither?
(c) Does it make more sense to expand f(x) in
a cos/sin or exponential Fourier series? Why?
(d) Expand f(x) in the most economical Fourier series. [Hint:
 xe
 inx
 1 ix 
dx  e  inx  2   .]
n
n
Boas 8.6.17 (p.423)
D
2
 4  y  16cos 4 x
(a)
(b)
(c)
(d)
Write the differential equation in terms of derivatives of y.
Solve for the roots of the characteristic (or homogeneous) equation.
Write the characteristic (or homogeneous) solution, yc, with undetermined coefficients.
Write the form of the particular solution, based on the nonhomogeneous part of the
equation.
(e) Check for constraints on undetermined coefficients.
(f) Write the general solution to the equation.
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How do you feel about your work and your learning so far?
What question do you wish had been on this exam?
Feedback or questions for your prof?
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