The following is a collection of questions that the instructor has given as either exam questions or 'sample exam' questions in the past.
This site is constantly under construction, to remove errors resulting in translating these questions from Microsoft Word format into html, and to compensate for the fact that some questions originally referred to Figures that are now absent.
2. Calculate the total amount of energy stored in the field of a sphere of charge Q and radius a, in which the charge is uniformly distributed inside the sphere.You will need to include the energy stored in the interior as well as the exterior. (Note: in one of the homeworks you calculated the energy stored outside the sphere equals kQ2/2a.
3. A slab of charge sits between z=-a and z=a. It is infinite
in extent in the x- and y-directions. Between a and -a, the charge density
is given by rho(z)=alpha z.
a) Find the electric field for z>a.
b) Find the electric field for 0<z<a
2 pi alpha a2k,2 pi alpha z2k
4. The electron in a He+ ion is, on average, 0.27A from
the center of the nucleus. Put the nucleus at the origin and the electron
along the positive x-axis.
a) What is the force on the electron if no other charges are present?
b) What is the magnitude of the total electric field at (x,y)=(0, 0.27A)?
c) What is the net flux through a box with 1 sides, centered at the origin?
5. At each corner of a square is a particle with charge q. Fixed at the center of the square is a point charge of opposite sign, of magnitude Q. What value must Q have to make the total force on each of the four particles zero?
6. A spherical volume of radius a is filled with charge of uniform density rho. We want to know the potential energy U of this sphere of charge, that is, the work done in assembling it. Calculate it by building the sphere up layer by layer, making use of the fact that the field outside a spherical distribution of charge is the same as if all the charge were at the center. Express the result in terms of the total charge Q in the sphere.
7. Imagine a sphere of radius a filled with negative charge of uniform density, the total charge being equivalent to that of two electrons. Imbed in this jelly of negative charge two protons and assume that in spite of their presence the negative charge distribution remains uniform. Where must the protons be located so that the force on each of them is zero? (This is actually the reverse of J. J. Thomson's plum pudding model of the atom, in which electrons were 'plums' hiding in the positive jelly of the atom. Theorists actually calculated the equilibrium configurations for quite a few of the elements in this model.)
8. A) A point charge q is located at the center of a cube of edge length d. What is the value of the integral of E-dot-da over one face of the cube?
B)The charge q is moved to one corner of the cube. What is now the value of the flux of E through each of the faces of the cube?
1. Four positive charges of magnitude Q form a square with coordinates (x,y)=(±a, ±a).
a) What is the electrostatic potential of another positive charge q at a position x along the x-axis?
b) From your results to part (a), show that the charge q would be at equilibrium at the origin.
2. Consider the vector function: F=-yi+xj+3k.
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.
1. The capacitance of two nested spherical shells, of radii a and b (b>a), is given by C=ab/(b-a).
a) Verify that the standard expression for the energy stored in the electric field is equal to the standard expression for the energy stored in a capacitor if the shells each have a charge Q on them.
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 (xo,yo) lying between the sheets.
a) State the location and quantity of all the image charges in terms of xo 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. Consider two capacitors in parallel with a potential
difference, V, across them, and an equivalent single capacitor with the
same voltage across it.
a) Show that the sum of the charges on two parallel capacitors is the same as the charge on the equivalent capacitor.
b) Show which system stores a larger potential energy, the parallel capacitors or the single capacitor.
4. A 100pF capacitor is charged to 100V. After the charging battery is disconnected, the capacitor is connected in parallel to another capacitor. If the final voltage is 30V, what is the capacitance of the second capacitor? How much energy was lost?
5. 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, and how much energy can be stored?
6. Coaxial cable consists of two concentric conducting cylinders of radii a and b (a<b). The cable can be characterized by its capacitance per unit length. If a length, L, of the inner cylinder contains a charge per length, lambda=Q/L,
a) Calculate the E-field between the two cylinders,
b) Calculate the potential difference between the two cylinders.
c) Show that the capacitance per length of the coax equals 1/(2ln(a/b)). (CGS)
d) Calculate the capacitance per meter (in MKS units) for "75ohm" cable, for which a=0.032" and b=0.139". Notice that capacitance per length is dimensionless in CGS, but has units of F/m in MKS units.
e) Verify that the formula for the energy stored in the E-field agrees with the formula for the energy stored in a capacitor, for this system.
7. A capacitor consists of a small spherical conductor of radius r near an infinite conducting plane, located a distance h away. The voltage difference between the two conductors is Q/r when charges of magnitude Q are placed on the two conductors.
a) Calculate the capacitance of this system and the total energy stored in the capacitor.
b) Calculate the potential difference, the capacitance, and the energy stored if this system is replaced by the sphere and its 'image charge'.
c) Why is there a factor of two discrepancy between the energy storage in the two systems? Explain in terms of the energy stored in an electric field.
1. Consider two circuits: In one, a resistor, R, is in series with an Emf, E, and the two are in parallel with another resistor, R. In the other circuit, the first resistor is replaced with R', and the parallel resistor is replaced with an emf, E'. The Thévenin Emf and resistance of the following two circuits are the same.
a) Calculate the Thévenin Emf and Thévenin resistance.
b) Calculate R' and E'.
2. In measuring the resistivity of a low-resistance sample
of material, one generally uses a four-probe technique to avoid the effects
of voltage drops along the wires of a two-wire ohmmeter. Consider four
resistors, RA, RB, RC, and RD, which form a loop. A current source is connected
to the two ends of resistor RA, and a voltmeter is connected across RC,
which is opposite the loop from RA.
a) What is the measured resistance, R=V/I, for the network, if the individual resistances are as marked?
b) An increase in which resistances will increase the measured resistance?
3. Calculate the Thévenin Emf and resistance of the following circuit: R1, R2, and R3 form an equilateral triangle. A battery is connected across the two ends of R1, and the voltage across R2 is measured.
4. Two wires of material of conductivities sigma1 and sigma2 are joined together. A current, I, passes between the two at the junction. Show that there will be a buildup of charge equal to (I/4pi)(1/sigma2-1/sigma1) at the junction.
5. A wire is drawn through a die, reducing its diameter by 25% and increasing its length. By what factor will its resistance be increased? Later it is flattened into a ribbon by rolling, which results in a further increase in its length, which is now twice the original length. What has been the overall change in resistance? Assume the density and resistivity remain constant throughout.
6. Two graphite rods are of equal length. One is a cylinder of radius a. The other is conical, tapering linearly from radius a at one end to radius b at the other. Show that the end-to-end electrical resistance of the conical rod is a/b times that of the cylindrical rod. Hint: Consider the rod made up of thin, disklike slices, all in series.
7. Three identical resistors of resistance R0 are arranged
to form a T. What value of resistance, R, must be attached to the network
(shown below) to make the effective resistance of the total network also
equal to R?
--- R0----------- R0-----
8. Consider a black box which is approximately a 10cm cube with two binding posts. Each of these terminals is connected by a wire to some external circuits. Otherwise the box is well insulated from everything. A current of approximately 1A blows through this circuit element. Suppose now that the current in and the current out differ by one part per million. About how long would it take, unless something else happens, for the box to rise in potential by 1kV? (You may wish to 'assume a spherical cube'.)
1. Fixed in the frame F is a sheet of charge, of uniform surface density sigma, which bisects the dihedral angle formed by the xy and the yz planes. The electric field of this stationary sheet is of course perpendicular to the sheet. Compare 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. What is the surface charge density sigma' in F'? Is the electric field perpendicular to the sheet in this frame?
2. 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 electric field, in CGS 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?
3. In the lab frame, two equal and opposite charges are
observed to approach each other along the x-axis at 0.6 times the speed
of light. At t=0, they collide elastically, and each is reflected by 180o.
a) If we draw 8 electric field lines converging on (diverging from) each point charge, what is the angle between the x-axis and the lines which are not along either the x- or y-axis?
b) Sketch the electric field lines near the two charges on the attached graph paper at t=5 seconds, using a scale of 1cm block = 1 light second, the distance light travels in one second. (You can ignore what the lines look like exactly 5 light seconds from the origin, but sketch the lines for distances greater than and less than 5 light seconds.)
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. 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 velcity, v, in the positive x-direction. Note that the force on q is the same for both frames; this may help you calculate one of the four quantities that is otherwise very difficult to calculate.
1. I am given a slab of material 2mm long with a 0.2mm square cross-sectional area. I attach current leads at the two ends 2mm apart. If I can place the sample in a 17kG magnetic field and send 100mA of current through it, what is the density of charge carriers in the unknown material if I measure a Hall voltage of 1 microV? If the mobility of the charge carriers is 1500cm2/V/s, what is its resistivity?
(The Hall voltage is given by VH=(RHIB)/t, with the Hall coefficient given by RH=1/(nqc) in CGS, or by RH=1/(nq) in MKS.)
2. A researcher wants to produce a Hall signal of at least
1microV, in order to get three significant figures in the measurement.
The available sample is 2cm long in the direction of current flow, 1cm
wide (distance between the Hall probes) and 1mm thick. The material's Hall
coefficient is consistent with it having a density of carriers of 1020/cm3.
There is a 20kG magnet available to the researcher.
a) How much current must the researcher apply to the sample in order to get a 1microV signal?
b) A 10mV resistive signal is registered between two electrodes 2cm apart in the direction of current. What is the resistivity of the material?
c) Which of the following quntities could be adjusted to ensure that the Hall signal is no less than 0.1% of the resistive voltage -- the current, the magnetic field, the thickness of the sample, the width, the current density?
1. A thin ring of radius a carries a static charge q. This ring sits in a magnetic field B which is parallel to the ring's axis, and the ring is supported so that it is free to rotate about that axis. If the field is suddenly switched off, show that it picks up an angular velocity which is independent of how long it took the field to go to zero.
2. Starting with the expression for the energy stored in an inductor, verify that, for an ideal (infinitely long) solenoid, the energy density is given by u=B2/8(pi).
3. Consider a magnetic potential function given by A=z3i+x2j-yx3k.
a) Calculate the magnetic field associated with this potential.
b) Verify that your answer to part (a) is a valid magnetic field function.
c) What current density is required to produce such a magnetic field?
4. A solenoid of 100mH self-inductance consists of an
80ohm resistance coil of wire. The solenoid has been connected to a 120VDC
power supply for a very long time.
a) How much energy is stored in the B-field of the inductor?
b) The solenoid is now disconnected from the power supply. How much Joule heating is dissipated by the solenoid in the first millisecond after the power supply is disconnected?
c) How much energy is dissipated by Joule heating by the time the current in the solenoid goes to zero?
5. I am given a slab of material 2mm long with a 0.2mm square cross-sectional area. I attach current leads at the two ends 2mm apart. If I can place the sample in a 17kG magnetic field and send 100mA of current through it, what is the density of charge carriers in the unknown material if I measure a Hall voltage of 1microV? If the mobility of the charge carriers is 1500cm2/V/s, what is its resistivity?
6. Given the electric field E=Pk×r, where P is some
constant, k=(0,0,1), and r=(x,y,z), at some point in time,
a) Could this describe an electrostatic field? Why or why not?
b) If not, what can you say about the magnetic field at this particular point in time?
7. An alternating current in a primary coil produces a
magnetic field of B0eiwt in the plane of a secondary coil of
radius 3cm. B0=30G, and the current is alternating at f=60Hz.
a) What is the maximum flux through the secondary coil?
b) What is the maximum induced Emf in the secondary?
c) If the two coils are identical and each consists of a single turn of wire, what current is needed in either coil to produce 30G at its own center?
8. 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 energy densities of the each field.
9. You have designed a solenoid for a cryogenic Hall experiment. The solenoid is 2cm long with 100 turns per cm. The cross section of the solenoid is circular, with a radius of 5mm.
a) Estimate the inductance of the solenoid, using the approximation that the magnetic field is nearly equal to the field inside an infinitely-long solenoid. Note 1H=1.113×10-12s2/cm
b) If you use copper wire (rho=3microOhm-cm) with a radius of 0.05cm, what is the resistance of the wire?
c) What voltage across the solenoid will give a steady-state 100G field?
d) You can think of the solenoid as a series combination of a resistor and an inductor. If the voltage in part (c) is suddenly reversed, what is the rate of change of the current?
1. The equations for the capacitance per length and the inductance per length, respectively, of a coax cable are [2ln(b/a)] -1 and 2ln(b/a)/c2 (CGS), given that the inner conductor has a radius of a, and the outer conductor has an inner radius of b. Lengths of coax with specific values of a and b are characterized by their capacitive or inductive impedance at resonance (the magnitudes of which are equal).
a) Show why this is a better way of characterizing the coax than, say, the resonant frequency, omega. In particular show that one of these quantities depends on the length of cable and the other one doesn't.
b) Calculate the impedance of 75Ohm cable in CGS units.
c) What can one say about the dimensions of 75Ohm cable?
2. A 2mOhm, 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) The voltage doesn't last, because charge collects at the ends of the wire, producing an opposing Emf. How many excess electrons build up on the negatively charged end?
3. The series combination of a resistor and inductor is
in parallel with a capacitor. The leads of the capacitor are attached to
a) What is the complex impedance across the posts?
b) Under what circumstances is it possible for that impedance to be purely real?
4. Coax cable can be thought of as a parallel combination
of resistance R, inductance L, and capacitance C.
a) Calculate the total (complex) impedance of this network at a frequency omega.
b) What is the resonant frequency for this network?
c) What is the impedance of the network at this frequency?
5. The amplification of an op amp inverting amplifier,
A=Vo/Vi, where Vi is the input voltage and Vo is the output voltage, is
equal to -Z2/Z1, where Z1 is the input impedance and Z2 is the "feedback"
impedance. If the feedback impedance is a capacitor, C, and the input impedance
is a series combination of an inductor and a resistor,
a) Calculate the complex amplification as a function of frequency, f. Be sure to put it in the form x+iy.
b) For what value[s] of omega is the amplification a purely real number?
6. A resistor, R, is in parallel with an inductor, L,
and the two of them, together, are in series with a capacitor, C.
a) Calculate the complex impedance of the circuit, as a function of the angular frequency, omega.
b) For what value of frequency, f, does the magnitude of the impedance reach its maximum value?
7. An inverting amplifier can be made from an op amp with an input resistor R1 and a feedback resistor R2. The amplification will equal A=-R2/R1. If either of these resistors is replaced with a capacitor, inductor, or some other combination of R, L, or C, the resistances in this expression should be replaced with impedances, Z.
a) Which resistor should be replaced by a capacitor to make a low-pass filter, and which should be replaced by a capacitor to make a high-pass filter? What is the amplification for each?
b) Claculate the frequency-dependence of the amplification if the input impedance is a 10k resistor, and the feedback impedance is a chose that behaves like a 50mH inductor in series with a 100ohm resistor. At what frequency is the magnitude of A equal to 0.01?
1. Two identical electric dipoles, p, sit along the x-axis, one at the origin, the other at x=+d. The dipole at the origin, p1, points at an angle of 45o above the positive x-axis, while p2 points at a direction of 135o.
a) Calculate the electric field at p2 due to p1, in terms of Er and Etheta.
b) Calculate the potential energy of the system of charges.
2. A water molecule is an electric dipole with dipole
moment p=1.84×10-18esu.cm. Consider a microwave oven which
deposits 500W into a 1L=(10cm) 3 cube of water.
a) If the power is uniformly deposited on the surface of the cube, what is Erms at the surface?
b) What is the average energy needed to flip a water molecule from an average random orientation U=0, to full alignment with the field?
c) If the whole 500J is absorbed by the water in one second in producing spin flips, with that energy quickly dissipated by collisions between molecules, and if the liter of water contains (1000/18)NA, where NA=6×1023 molecules, then how many spin flips, on average, does each molecule undergo in one second?
3. Two identical electric dipoles, p, are a distance d
apart, with the following orientations: p1 is on the left and points up; p2 on the right, points
a) Calculate the torque on p2 due to p1.
b) Is this torque clockwise or counterclockwise?
c) Is the torque on p1 due to p2 clockwise or counterclockwise?
4. The index of refraction in air at room temperature is about 1.00027 for visible light.
a) Claculate the corresponding dielectric constant.
b) Calculate the electric susceptibility.
c) The susceptibility is just the polarizability per volume. What is the polarizability of each molecule in air if one mole occupies approximately 22L? The units of polarizability should be units of volume, and should approximate the volume of the individual molecule.
5. Estimate the capacitance of a 'Leyden jar', a bottle which is covered on the outside by foil and which contains water. The water acts as one of the two electrodes. Make some reasonable assumptions about the bottle's dimensions. The dielectric constant of glass can be assumed to be about 4. Take the thickness of the glass to be about 2mm.
6. Consider two electric dipoles sitting along the x-axis, separated by r=3A. If the dipoles begin pointing in parallel, how much energy does it take to flip one of them, if
a) they start out pointing in the x-direction?
b) they start out pointing in the y-direction?
Calculate the dipole moment necessary for the energy in part (a) to equal kT=4.0×10-14erg. If the thermal energy is smaller than the energy above, thermal agitation will be modest, and the dipoles can stay aligned.
7. Calculate the force on a dipole near a point charge, +Q, located at the origin. The dipole is located on the x-axis, is poining in the positive x-direction, and has a magnitude p. (Be sure you can convince yourself that the form of the answer -- which is not zero -- makes sense.)
1. Magnets are often characterized by their 'lift', the amount of force they can pull. Assuming that you are using a magnet to pull some iron, and assuming that you induce a dipole moment in the iron equivalent to the dipole moment of the magnet, derive an expression for the lift of a spherical magnet of radius r, in terms of its magnetization, M.
2. Show that an electromagnetic wave of the form
satisfies Maxwell's equations in their final form (inside a material). What is the relation between Eo and Bo? Between k and w?
3. The index of refraction actually equals sqrt(mu*epsilon) for a material with nontrivial electric and magnetic properties. The quantity mu is called the magnetic permeability, and B=mu*H. Given that the parallel component of H is continuous across an interface, consider an electromagnetic wave going from a vacuum (where epsilon=mu=1) into another medium, with H pointing into the paper. Apply the appropriate boundary conditions on E, D, and H, as well as the law of reflection and Snell's law, to calculate the ratio of H to E in the refracting material.
1.1: kQr2/a4, kQ/r2
1.4: 0.63microN, 3.3×1012V/m, 18nVm
1.8: 2(pi)q/3. zero and (pi)q/6
2.2: Cannot, 2k
3.2: (4 pos., 4 neg.), Along the sheets. Yup.
4.1: E'=E/2, R'=0.75R.
5.1: E'=1.13E. E points at -45o and 135o. E' at -51o and 129o.
6.1: 5.3×1027/m3. 0.8microOhm-cm
6.2: 8mA. 0.063ohm-cm. B
8.1: Omega=c/l, XL=(2/c)ln(b/a), 8.3×10-11s/cm,
10.1: sqrt(2)p/d3, sqrt(2)p/2d3, -p2/2d3
11.2: kc=sqrt(mu*epsilon)w, Bo=sqrt(mu*epsilon)Eo