Download PHYS 308

PHYS 308
Spring 2004
________________________
Daniel W. Koon
SAMPLE EXAM I
(Be sure you know your prefixes from pico through mega.)
Useful constants: k=9×109N·m2/C2, e=1.602×10-19C
1. Three equal charges of +2nC sit in the xy-plane. One is at the origin, another on the y-axis at y=+2cm,
and a third on the x-axis at x= -3cm.
a) What is the potential energy of this collection of charges?
b) What is the magnitude of the force experienced by the charge at the origin?
2. The charge distribution inside a sphere of charge Q and radius a is given by  (r )  Qr
zero outside.
a) Show that the total charge of the sphere is Q.
b) What is the electric field outside the sphere? (r>a)
c) What is the electric field inside the sphere? (r<a)
3. Given that V  G ( x  y ) / 4 0 in some region of space,
2
2
a) Find the density of charge.
b) Find the electric field.
c) Sketch an appropriate Gaussian surface for this problem.
a 4
inside, and
PHYS 308
Spring 2004
________________________
Daniel W. Koon
EXAM I
(Be sure you know your prefixes from pico through mega.)
Useful constants: k=9×109N·m2/C2, e=1.602×10-19C
1. Verify that the electric field is the gradient of the potential for the following case. Consider an electron
at the origin.
a) What is the potential for a point at x=0.99m along the x-axis? For x=1.01m? Keep at least five
significant figures for each.
b) Estimate Ex from the results of part (a), to four significant figures.
c) Calculate the electric field at x=1.00m directly, keeping four significant figures.

3
2
2. Given the electric field, E  Ar rˆ  Ar ( xiˆ  yˆj  zkˆ) , within a sphere of radius R.
a) Show that the charge density, , in this region of space is given by
want to use the fact that
r
x
  5 Ar 2 /( 4k ) . (You may
 rx , etc.)
b) Show that the total charge inside a sphere of radius R is Q
c) Verify Gauss’ Law for a sphere of radius R.
 Ar 5 / k .

3. Consider the vector function: F  2 xyiˆ  x 2 ˆj  3kˆ .
a) Can this function represent an actual electrostatic field? Why or why not?
b) If the function can represent an electrostatic field, find the electrostatic potential function.
Otherwise, calculate its curl.
PHYS 308
Spring 2005
________________________
Daniel W. Koon
SAMPLE EXAM I
(Be sure you know your prefixes from pico through mega.)
Useful constants: k=9×109N·m2/C2, e=1.602×10-19C
1. The electric field a distance, z, above the midpoint of a uniform linear distribution of charge, of length
2L and charge density =Q/(2L) is given by
E
2kL
z z 2  L2

. Use this result to show that the E -field a
distance, z, above the center of a square formed by four such bars, each of length a, is given by
E
z
4akz
2
 (a / 2) 2

z2  a2 / 2
2. Given E x   A sin x,
E y  0  E z , show whether this vector can represent an electrostatic field.
(HINT: Look for an easier way to do this.) If so, then calculate an appropriate electric potential, V, and
charge density function, .

3
2
3. Given the electric field, E  Ar rˆ  Ar ( xiˆ  yˆj  zkˆ) ,
a) Show that the charge density, , in this region of space is given by
to use the fact that
r
x
  5 Ar 2 /( 4k ) . (You may want
 rx , etc.)
b) Calculate the total charge inside a sphere of radius R.
c) Verify Gauss’ Law for this electric field, for a sphere of radius R.
4. Consider a sphere of charge Q and radius a, in which the charge is uniformly distributed inside the
sphere.
a) Show that the electric field inside is kQr/a3.
b) Show that the total amount of energy stored in the electric field is 3kQ2/5a. You will need to include the
energy stored inside as well as outside the sphere.
PHYS 308
Spring 2005
________________________
Daniel W. Koon
EXAM I
(Be sure you know your prefixes from pico through mega.)
Useful constants: k=9×109N·m2/C2, e=1.602×10-19C
1. Consider three charges on the z-axis:
q1  2q at z   d ,
q 2  2q at z  d ,
q3  3q at z  3d .
a) Calculate the magnitude and direction of the electric field on the z-axis at z=+3d.
b) Calculate the electric potential at the same location.
2. Given
E x  2 yz  z 

 , provide an appropriate Ez that would make this a valid electrostatic field.
E


2
xz
y


Calculate an appropriate electric potential, V, and calculate the charge density function, .
3. The charge distribution inside a sphere of charge Q and radius, a, is given by  (r )  Qr / a 4 inside, and
zero outside.
a) Show that the total charge of the sphere is Q.
b) What is the electric field outside the sphere? (r > a)
c) What is the electric field inside the sphere? (r < a)
4. Designate the corners of a square, 5cm to a side, in clockwise order, A, B, C, and D. Put a charge 2nC

at A, +5nC at B. Determine the value of the line integral of the electric field, E , from point C to point D,
without doing the actual integral. (I will not give you full credit for doing the integral.)
PHYS 308
Spring 2004
________________________
Daniel W. Koon
SAMPLE EXAM II
(Be sure you know your prefixes from pico through mega.)
Useful constants: k=9×109N·m2/C2, e=1.602×10-19C
P0(x)=1; P1(x)=x; P2(x)=(3x2-1)/2; P3(x)=(5x3-3x)/2
1. Consider a boundary-value problem consisting of a square plate with +100V on two adjacent sides and
-100V on the other two. We are going to model this continuous system with the following discrete system.
Calculate the electrical potential at all of the internal points.
+100V
+100V
+100V
+100V
0V
+100V
+100V
+100V
-100V
-100V
-100V
0V
-100V
-100V
-100V
-100V
2. Two point charge lie on the positive z-axis above an infinite conducting xy-plane. Charge 1 is a charge
of +q located at z=3d. Charge 2 is a charge of -2q at a position of z=d. Calculate the electrostatic potential
in the half-plane above the conducting surface.
3. The capacitance of two long concentric cylinders of length L and radii a and b (L>>b>a), is given by
C=L/[2kln(b/a)].
a) Verify this expression.
b) As d=b-a approaches zero, show that this expression approaches the value for a parallel plate
capacitor of thickness d and plate area, A, equal to the surface area of either cylinder. (Hint: ln(1+x) = x x2/2 + x3/3 + ...)
c) Show that the total energy stored in this system equals Q 2/2C.
4. A point dipole at the origin (x,y,z)=(0,0,0) points in the positive z direction.
a) What are the x- and z-coordinates of the E-field due to this dipole at the position (d,0,d)?
b) Put an identical dipole at the point (d,0,d), also pointing “up”. What is its potential energy?
PHYS 308
Spring 2004
________________________
Daniel W. Koon
EXAM II
(Be sure you know your prefixes from pico through mega.)
Useful constants: k=9×109N·m2/C2, e=1.602×10-19C
P0(x)=1; P1(x)=x; P2(x)=(3x2-1)/2; P3(x)=(5x3-3x)/2
1. Consider a 66 lattice being used to solve the following boundary-value problem of Laplace’s equation.
a) Using symmetry, how many different independent unknowns remain to be solved? (e.g. If one cell is
+A and another is -A, where A is some unknown constant, these two would count as a single independent
unknown.)
b) Write equations for each of these independent variables.
c) Given that the missing value in the top left is approximately 82V, fill in the rest of the lattice.
+100V
+100V
+100V
+100V
+100V
0
+100V
-100V
+100V
-100V
+100V
-100V
+100V
-100V
0
-100V
-100V
-100V
-100V
-100V
2. The electrostatic potential on a sphere of radius R is given by V(R,) = Asin2.
a) Express V(R,) as a sum of Legendre polynomials.
b) Find an expression for the electrostatic potential V(r,) for r<R.
3. The capacitance of two nested spherical shells, of radii a and b (b>a), is given by C=ab/k(b-a).
a) Verify this expression.
b) As d=b-a approaches zero, show that this expression approaches the value for a parallel plate
capacitor of thickness d and plate area equal to the surface area, A, of either sphere.
c) Verify that, for this system, the total energy stored equals Q2/2C.
4. Four dipoles sit at the corner of a square in the xz plane with a distance d between nearest neighbors.
Each dipole has a magnitude p. The two on the left point up, the two on the right point down.
Consider the point at the center of the square.
a) What are the x and z components of the E field at the point at this point in the center, due to the dipole
at the top left corner?
b) What is the total E field at the center, due to all four dipoles?
PHYS 308
Spring 2005
________________________
Daniel W. Koon
SAMPLE EXAM II
(Be sure you know your prefixes from pico through mega.)
Useful constants: k=9×109N·m2/C2, e=1.602×10-19C
P0(x)=1; P1(x)=x; P2(x)=(3x2-1)/2; P3(x)=(5x3-3x)/2
1. Consider a boundary-value problem consisting of a
square plate with +100V on two adjacent sides and -100V
on the other two. We are going to model this continuous
system with the following discrete system. Calculate the
electrical potential at all of the internal points.
+100V
+100V
+100V
+100V
0V
+100V
+100V
+100V
-100V
-100V
-100V
0V
-100V
-100V
-100V
-100V
2. The electrostatic potential on a sphere of radius R is given by V(R,) = Asin2.
a) Express V(R,) as a sum of Legendre polynomials. Show that V ( R,  ) 
b) Find an expression for the electrostatic potential V(r,) for r<R.
2A
( P0  P2 )
3
3. Four dipoles sit at the corner of a square in the xz plane with a distance d between nearest neighbors.
Each dipole has a magnitude p. The two on the left point up, the two on the right point down.
Consider the point at the center of the square.
What are the x- and z- components of the contribution of the top left dipole to the E field at the center?
4. A 100pF parallel-plate capacitor is charged to 100V. After the charging battery is disconnected, a
dielectric slab with dielectric constant 80 is inserted between the plates of the capacitor. What is the final
voltage, charge, and capacitance of the system? How much energy was lost by the electric field?
PHYS 308
Spring 2005
________________________
Daniel W. Koon
EXAM II
(Be sure you know your prefixes from pico through mega.)
Useful constants: k=9×109N·m2/C2, e=1.602×10-19C
P0(x)=1; P1(x)=x; P2(x)=(3x2-1)/2; P3(x)=(5x3-3x)/2
1. The following is a very crude Relaxation Method exercise for solving
Laplace’s Equation. Let’s impose the following boundary conditions: Set the top
A
at +100V and the right hand side at -100V (Dirichlet conditions), and the
E
normal electric field equal to zero on the lefthand and bottom sides. (Neumann
I
conditions) Let the letters A, B, C, etc. stand for the potential at each gridpoint.
M
We would then write the following equations:
A = B = C= +100V; H = L = P = -100V; D = (C+H)/2
Write out the equations for the electric potential in each of the following cells: M, I, N, G.
B
F
J
N
C
G
K
O
D
H
L
P
2. Two metal sheets intersect at a 45o angle at the origin. One lies along the x-axis and the other lies
along the line x=y. (All sheets are parallel to the z-axis.) Consider a point charge +Q at the point (x o,yo)
lying between the sheets.
a) State the location and quantity of all the image charges in terms of x o and yo, sketching their location
for xo=4, yo=1.
b) Where does the electric potential equals zero? Is this consistent with it vanishing at infinity as well?
Why?
3. A 'Leyden jar' is a bottle which contains water and which is covered on the outside by foil. The water
acts as one of the two electrodes.
(a) Calculate the capacitance of a Leyden jar of diameter 30cm, with the water level 10cm high, and with
0.2cm thick glass walls, with dielectric constant of 4.
(b) How much charge is stored on either electrode if we place 3kV across the jar?
4. We want to design a spherical vacuum capacitor with a given radius a for its outer sphere, which will be
able to store the greatest amount of electrical energy subject to the constraint that the electric field
strength at the surface of the inner sphere may not exceed E0. What radius b should be chosen for the
inner spherical conductor? ( C

ab
for nested spherical capacitors.)
k ( a  b)
PHYS 308
Spring 2004
________________________
Daniel W. Koon
SAMPLE EXAM III
c=3×108m/s, e=1.602×10-19C
k=9×109N·m2/C2, 0=1/(4k), 0=4×10-7C2/N·m2


1. Can E  s  kˆ  yiˆ  xˆj represent an electrostatic field? If yes, calculate the electric potential. If no,
calculate a possible expression for the magnetic field.
2. A wire on the x axis carries a current I in the positive x direction. A square of length a is in the xy plane
with its nearest edge at y=+s (its furthest edge at y=s+a).
a) What is the magnetic flux in the loop?
b) If the square is pulled from the wire at a speed v, what is the induced Emf in the loop? What is its
direction?
3. Consider a current-carrying slab, of thickness a, which carries a current density of J = I / A.
a) Place the slab on the xy plane, with a/2 of its thickness above the plane, and a/2 of it below. Calculate
the magnetic field along the z-axis as a function of z, both inside and outside the slab.
b) Show the direction of the magnetic field on a diagram, clearly demonstrating its direction above, below,
and inside the slab. Be sure to show the direction of the current as well.
4. Your instructor is studying dysprosium, a material in which the negative Hall coefficient seems to go to
zero, and possibly change signs as it absorbs hydrogen. Its resistivity is about 100 ·cm.
a) Why does a zero Hall coefficient suggest that our interpretation of how the Hall coefficient is related to
microscopic quantities inside a material may be a tad too simplistic?
b) If a dysprosium specimen has a thickness of 17nm (just barely enough to ensure that it completely
covers the material underneath) and a width and length of 0.5cm each, if we can push 3mA of current
through it at a field of 0.6T, and if we measure the Hall voltage to be less than 10nV (about the limits of
our instruments), then what is the upper level for the Hall coefficient of partly hydrided dysprosium? (For
comparison, note that the Hall coefficient of most metals is about 10 -10C/m3.)
c) What is the resistance of the specimen, if we treat it as a wire?
PHYS 308
Spring 2004
________________________
Daniel W. Koon
EXAM III
c=3×108m/s, e=1.602×10-19C
k=9×109N·m2/C2, 0=1/(4k), 0=4×10-7C2/N·m2

1. Can B  skˆ 
x 2  y 2 kˆ represent a valid magnetostatic field? (That is, can both electric and
magnetic fields be constant for this field?) If so, calculate the current density function. If not, calculate the
magnetic potential function.

2. The magnetic field in a region of space is given by B  Aykˆ sin t . Calculate the Emf generated
around a square loop in the xy plane with one corner at the origin, and its opposite corner at (x,y)=(a,a).
What is the direction of the induced current at t=0? Is it clockwise or counterclockwise in the xy plane?
3. Consider a wire of radius a with a uniform current density, J = I / a2.
a) Calculate the magnetic field inside the wire as a function of radius, s<a, from the center axis, in terms
of the total current, I.
b) Calculate the magnetic field outside the wire as a function of the radius, s>a, also in terms of I.
c) What is the direction of the force acting on the current inside the wire?
4. The self-inductance of a solenoid is given in MKS units as L  0 n lr , where n is the number of
2
2
windings per length, l, of the length of the solenoid, and r is the radius of the solenoid. The magnetic field
in the center of an infinitely long solenoid is 0nI, where I is the current. Calculate the energy stored in a
solenoid two ways:
a) by calculating the energy density at the center of the solenoid (and assuming it is uniform throughout),
and
b) by calculating the energy stored in an inductor (also assuming a uniform B).
5. Consider four infinitely long wires each carrying 3A. Looked at from the side, the four wires form the
corners of a square of 25 cm to a side. If we look at the wires from the side, and if the top two carry
current into the page, and the bottom two carry current out of the page, what is the magnitude and
direction of the total magnetic field at the point between and equidistant from the wires?
If we put a wire at that location carrying current out of the plane, what would be the direction of the force
acting on it?
5A. Three long straight parallel wires are
located as shown in the diagram. One wire
carries current 2I into the paper; each of the
others carries current I in the opposite
direction. What is the strength of the
magnetic field at the point P1 in the center of
the figture?
PHYS 308
Spring 2005
________________________
Daniel W. Koon
SAMPLE EXAM III
(Be sure you know your prefixes from pico through mega.)
c=3×108m/s, e=1.602×10-19C
9
k=9×10 N·m2/C2, 0=1/(4k), 0=4×10-7N/A2
1. Consider a 10nm thick film of palladium [Pd] covering a 75mm  25mm microscope slide (approx.
1”3”). The two 25mm ends are covered with a much thicker metallic film, effectively shorting them and
ensuring that the film behaves like a long thin wire. The accepted value of the resistivity and Hall
coefficient of Pd are 11·cm and -0.810-10m3/C.
a) Calculate the resistance of the film and the power dissipation in the film if a 3mA current passes
through it.
b) How large a magnetic field must be applied perpendicular to the plane of the film to get a ±1V Hall
voltage?
c) If the Hall electrodes are not properly aligned, there will be a misalignment voltage that can be much
larger than the Hall voltage. How much must the electrodes be misaligned (along the length of the film) in
order to get a 1V voltage from this effect?
(Hint: you can calculate the current density, then the electric field, then the distance that will have a 1V
voltage along its length.)
2. Given that the kinetic energy of a [511keV/c2] electron is 13.6eV (The potential energy is -27.2eV, so
the total energy is -13.6eV.),
a) Show that the electron’s [nonrelativistic] speed is 0.0073c = 2.2Mm/s if its radius is 0.53Å.
b) Calculate the magnetic field that the electron’s motion produces at the location of the nucleus. Notice
that this leads to splitting of the nuclear states, depending on whether the nuclear magnetic moments
align with this field or not. (In other words, this effect should be observable experimentally.)


3. Can E  s  kˆ  yiˆ  xˆj represent an electrostatic field? If yes, calculate the electric potential. If no,
calculate a possible expression for the magnetic field.
4. Consider a current-carrying slab, of thickness a, which carries a current density of K = I / A into the
plane of your paper.
a) Place the slab on the xy plane, with a/2 of its thickness above the plane, and a/2 of it below. Calculate
the magnetic field along the z-axis as a function of z, both inside and outside the slab.
b) Show the direction of the magnetic field on a diagram, clearly demonstrating its direction above, below,
and inside the slab. Be sure to show the direction of the current as well.
PHYS 308
Spring 2005
________________________
Daniel W. Koon
EXAM III
(Be sure you know your prefixes from pico through mega.)
c=3×108m/s, e=1.602×10-19C
9
k=9×10 N·m2/C2, 0=1/(4k), 0=4×10-7N/A2
1. The resistivity and Hall coefficient are 1.7·cm and -0.5510-10m3/C for copper, and 4500·m
[different units] and 625m3/C for pure silicon at room temperature.
a) Calculate the density of charge carriers and their mobility for both materials. (Clearly label which is
which.) Express the units of mobility as an appropriate combination of meters, volts, and seconds.
b) Compare the two densities and the two mobilities and explain briefly why one material has a greater
density and the other has a larger mobility.
2. Picture four infinitely-long current-carrying wires perpendicular to the plane of this page, arranged in a
square as below. The distance between any two nearest neighbors is 1cm.
a) If the top left wire carries 3A pointing into the page, what is the magnitude and direction of the field it
produces in the center of this square?
b) In order for the net magnetic field at the center of the square to point to the left of the page, show the
directions of the currents in the remaining three wires.
c) What is B/Bo -- the magnitude of this field, relative to the field in part (a) -- for this configuration?
d) If we place a fifth wire, of 1m length, in the center of this square, what is the direction and magnitude of
the current that would have to flow through it for the other wires to counteract its 12N weight so as to
levitate it?

3. Can B  skˆ 
O
O
O
O
x 2  y 2 kˆ represent a valid magnetostatic field? (That is, can both electric and
magnetic fields be constant for this field?) If so, calculate the current density function. If not, calculate the
magnetic potential function.
4. Consider a wire of radius a, with a uniform current density, J = I / a2.
a) Calculate the magnetic field inside the wire as a function of radius, s<a, from the center axis, in terms
of the total current, I.
b) Calculate the magnetic field outside the wire as a function of the radius, s>a, also in terms of I.
c) What is the direction of the force acting on the current inside the wire?
Physics 308
Daniel W. Koon
Spring 2005
Final Exam
c=(00)-1/2 = 3×108m/s, e = 1.602×10-19C
 0 = 410-7N/A2, 0 = 8.85x10-12F/m
1. A 2m, 1cm length of wire, of 0.001mm2 cross-section, is passed through a perpendicular 3T magnetic
field fast enough to produce an instantaneous Emf of 1.1V.
a) How fast was the wire dragged through the field?
b) Given that the electric field inside the wire is related to the charge density at the edges by E=4 k
=4kQ/A, calculate the excess charge on either end of the wire.
c) What is the total energy dissipated by the flow of charge in this rod? (If you prefer, you can calculate
this by calculating the energy density in the E-field inside the wire.)
2. Find the magnetic field associated with the electric field with these MKS coordinates:
Ex = 0
Ey = 0
Ez = Eo coskx cos ky cost
3. Consider the contents of a 700W microwave oven consisting of a cube 0.4m to a side.
a) Show that, if we remove the walls of the oven, radiation would leave the oven at a rate of about
700W/m2, assuming that it bounces equally off all the walls.
b) Calculate the energy density inside the oven.
c) Show that the rms electric field inside the oven is 520 V/m.
d) Calculate the total force (pressure times area) exerted by the radiation on the six oven walls.
4. A 10cm×10cm metal sheet lies in the xz-plane. It has a charge of 107 excess electrons on it.
a) In the frame in which the sheet is at rest, what is the magnitude and direction of the electric field, in
MKS units, just above the sheet?
What is the field at the same location if the sheet is traveling at 0.8c...
b) in the x-direction?
c) in the y-direction?
d) in the z-direction?
What is the electric charge on the sheet in each of these frames?
Physics 308
Daniel W. Koon
Spring 2005
REVISED Sample Final Exam
c=(00)-1/2 = 3×108m/s, e = 1.602×10-19C
 0 = 410-7N/A2, 0 = 8.85x10-12F/m
1. Verify that for an infinitely long solenoid, the expression for the energy stored in an inductor, L, carrying
a current, I, is equal to the energy stored in the magnetic field, where the energy density is given by
u=B2/2.
2. Given the equivalence of the two pre-Maxwellian forms of Ampere’s law
 
 

B
and
,
  B  0 J
  dl   0 I enclosed
 
 
develop an expression for  B  dl in terms of the electric flux,  e   E  da , if there are no free
 
currents present (J=0). (Hint: Consider how     E  dl depends on the magnetic flux, and compare
this to the corresponding Maxwell’s equation.)
3. Approximate the electric field inside the atom at the Bohr radius (0.53A) due to the hydrogen nucleus
(assuming the classical picture in which the electron is at the radius 100 percent of the time. Approximate
the magnetic field at the nucleus caused by the orbit of the electron (v=2.2Mm/s). Compare the maximum
energy density of the each field.
4. Two charges lie on the z-axis at t=0, one of them (q) at the origin, and the other (Q) at z=r. They
remain at the same positions relative to each other.
a) Give the magnitude and direction for both the electric field and the magnetic field at the origin due to Q,
for both the frame in which the two charges are at rest, and for a frame in which they both move at a

velocity, v , in the positive x-direction. You will need to calculate both the electric and the magnetic field in
each frame. Note: we didn’t discuss this in class, but the magnetic field in the second case is given by the
Lorentz transformation:
 


vE
B  B   2
c

[Okay: For the purists out there -- everyone else may safely ignore this note -- this v is the motion of the

two charges in this second frame. If v represents the relative motion of the two frames, then you need to
reverse the sign. That is why you’ll find a negative sign for this expression in textbooks.]
b) Find the force in both frames. Show that the force in the second frame is smaller than in the first frame
by a factor of 1 /  .
Physics 308
Daniel W. Koon
Spring 2004
Sample Final Exam
c=(00)-1/2 = 3×108m/s, e = 1.602×10-19C
 0 = 410-7N/A2, 0 = 8.85x10-12F/m
1. The electric field in some region of free space is


E  E0 i sin k z  ct  .
a) What is the corresponding magnetic field?
b) What is the magnitude and direction of energy flow (assume MKS units)?
2. Show that an electromagnetic wave of the form

E  E0 ˆj sin( kx  t )

B  B0 kˆ sin( kx  t )
satisfies Maxwell's equations in free space. What is the relation between E0 and B0? Between k and ?
3. The background radiation from the Big Bang, 14 billion years ago (14x109yr) is estimated at about
4x10-20J/cc (cubic centimeter), distributed (let’s assume) uniformly through space. You should imagine the
Universe to be a sphere of radius 14 billion light years (1 lt.yr = the distance light travels in one year).
a) What is the resultant [rms] electric field magnitude that we would observe for the background
microwave radiation?
b) What is the intensity (in W/m2) of the remnants of the Big Bang? (You can compare this to the Sun’s
1300W/ m2 at the top of the Earth’s atmosphere.)
c) What was the total energy radiated in the Big Bang? (This is a very crude estimate.)
4. Fixed in the frame F is a sheet of charge, of uniform surface density , which bisects the angle
between the xy and the yz planes. The electric field in frame F of this stationary sheet is of course
perpendicular to the sheet.
a) Calculate the magnitude and direction of the field as measured by observers in a frame F' that is
moving in the x direction with velocity 0.6c, to that measured by observers in frame F.
b) Is the electric field perpendicular to the sheet in this frame?
c) What is the surface charge density ’ in F'?
Physics 308
Daniel W. Koon
Spring 2004
Final Exam
c=(00)-1/2 = 3×108m/s, e = 1.602×10-19C
 0 = 410-7N/A2, 0 = 8.85x10-12F/m
1. Find the magnetic field associated with the electric field with these MKS coordinates:
Ex = 0
Ey = 0
Ez = Eo coskx cos ky cos t
2. Here is a particular electromagnetic field in free space, given in MKS units.
Ex = 0
Ey = Eo sin(kx+ t)
Ez = 0
Bx = 0
By = 0
Bz = -Bo sin (kx+ t)
Show that this field can satisfy Maxwell’s equations if  and k are related in a certain way. What is the
relation between Eo and Bo?
3. Consider the contents of a 700W microwave oven consisting of a cube 0.4m to a side.
a) Show that, if we remove the walls of the oven, radation would leave the oven at a rate of about
700W/m2.
b) Calculate the energy density inside the oven.
c) Show that the rms electric field inside the oven is 30mV/m.
d) Calculate the total force exerted by the radiation on the six oven walls.
4. A 10cm×10cm metal sheet lies in the xz-plane. It has a charge of 107 excess electrons on it.
a) In the frame in which the sheet is at rest, what is the magnitude and direction of the electric field, in
MKS units, just above the sheet?
What is the field at the same location if the sheet is travelling at 0.8c...
b) in the x-direction?
c) in the y-direction?
d) in the z-direction?
What is the number of excess electrons on the sheet in each of these frames?
QUESTIONS I WOULDA ASKED, BUT RAN OUTTA SPACE FOR:
4B. Two charges lie on the z-axis at t=0, one of them (q) at the origin, and the other (Q) at z=r. They
remain at the same positions relative to each other.
a) Give an expression for both the electric field and the magnetic field at the origin due to Q, for both the
frame in which the two charges are at rest, and for a frame in which they both move at a velocity, v, in the
positive x-direction.
b) Show that the force q at the origin is the same for both frames.
3. Calculate the potential energy , in CGS, of each of the following four pairs of magnetic dipoles, m,
separated by a distance b.


 
 
 
4. In nuclear magnetic resonance (NMR), photons having the appropriate frequency are absorbed by
nuclei for which the energy of the photon exactly equals the energy needed to flip the spin of the nucleus.
For the hydrogen nucleus, this magnetic moment equals 1.410-24J/T.
a) What frequency photons would we need in order to do NMR in our 1.7T magnet?
b) What is the frequency of a photon which would be absorbed via NMR in a 50T magnetic field (Earth’s
magnetic field)? (Useful constant: h=6.6310-34 J-s)
4. The Sun’s light has an intensity of about 1300W/m 2 at the top of the Earth’s atmosphere, 1.5x1011m
away from the Sun’s center.
a) What is the [rms] magnitude of the electric field at the Earth’s orbit?
b) What is the energy density of the electromagnetic radiation?
c) What is the total rate of power emitted by the Sun?
d) Write an expression for the magnitude of the electric field of this radiation as a function of distance, r,
from the center of the Sun. (For r>7x108m)
e) Is the integral of the energy density of the Sun’s radiation through all space bounded? (Explain or show
mathematically.)
1. Consider an infinitely long solenoid of radius R, with n coils per length. Inside the solenoid is a single
coil, coaxial with it, also of radius R. One can define the mutual inductance of the two as M12=2/I1, that
is, the mutual inductance is equal to the flux through the single coil divided by the current in the solenoid.
The interesting thing is that the mutual inductance is symmetric: it is also equal to the flux through the
solenoid divided by the current in the single loop. (ignoring all self-inductance fluxes)
a) Calculate the mutual inductance by calculating the flux through the single coil due to current in the
solenoid.
b) Show that this is equal to the mutual inductance due to current in the single coil. You do not need to
know the exact expression for the B-field along the axis of the coil, but Ampere’s law might come in
handy.
2. Given the equivalence of the two pre-Maxwellian forms of Ampere’s law

 
 
  0 J and  B  dl   0 I enclosed ), develop an expression for  B  dl in terms of the electric
 
flux,  e   E  da , if there are no free currents present (J=0). (Hint: Consider how
 
   E  dl depends on the magnetic flux, and compare this to the corresponding Maxwell’s equation.)


( B
3. Consider two electric dipoles, p, sitting a distance d from each other and oriented as shown to the left.
Calculate the torque, in CGS, on each dipole due to the electric field produced by the other.


4. A proton’s magnetic moment is 1.4110-24 J/T. Using a highly classical model in which this is the result
of a proton (charge = 4.810-10esu) with all of its charge at a radius of 1F (10-13cm) spinning about its
axis, calculate
a) the [circular] current caused by the proton’s rotation,
b) the tangential velocity at the edge of the proton.
1. A 2m, 1cm length of wire, of 0.001mm2 cross-section, is passed through a perpendicular 3T magnetic
field fast enough to produce an instantaneous Emf of 1.1V.
a) How fast was the wire dragged through the field?
b) Given that the electric field inside the wire is related to the charge density at the edges by
E=4k=4kQ/A, calculate the excess charge on either end of the wire.
c) What is the total energy dissipated by the flow of charge in this rod?
2. Consider a 1L glass bottle with 2mm thick walls, filled with water and covered with aluminum foil on its
outside. Making appropriate guesstimates for the dimensions of the bottle, and noting that the dielectric
constant of glass is 4, calculate the dielectric constant of this “Leyden jar”.
3. Consider an electromagnetic wave at the interface between two transparent materials, with the electric
field in the plane of the page, as shown below. Determine which boundary condition (e.g. matching



parallel or perpendicular components of the vectors E , D , B , or
below:
a)
E12 c
4

S ) is the source of each equation
cos  
E32 c
4
cos  
cE22 c
4n
cos 
b) E1 / c  E3 / c  nE 2 / c
c) E1 cos   E3 cos   E 2 cos 
d ) E1 sin   E3 sin   E 2 sin 