## Electromagnetism

- Electrostatics - Charge and Coulomb's Law
- Circuits
- Magnetism
- Biot-Savart Law

## What is an Electric Field?

## Coulomb's Law

## Gauss's Law

### Problem 1

A resistor is made from a hollow cylinder of length l, inner radius a, and outer radius b. The region a < r < b is filled with material of resisitivy ρ. Find the resistance R of this component

*R = ρ l / π b*^{2}*R = ρ l / π*(*b*^{2}-*a*^{2})*R = ρ l / π a*^{2}*R = ρ π b*^{2}/*l**R = ρ π*(*b*^{2}-*a*^{2}) /*l*

### Solution 1

I knew the answer would include *b*^{2} - *a*^{2}, so **B** and **E** would be good guesses. To start, get the current density:

### Problem 2

What is the Hall Effect?

### Solution 2

The generation of a voltage when a current carrying conductor is placed in a magnetic field.

If a magnetic field is applied to a current carrying conductor perpendicularly, an electrical potential difference is generated. This physical phenomenon was discovered by E.H. Hall in 1879 and consequently is known as the Hall effect

### Problem 3

An infinitely long, straight wire carrying current *I*_{1} passes through the center of a circular loop of wire carrying current *I*_{2}. The long wire is perpendicular to the plane of the loop. Which of the following describes the magnetic force on the loop?

- There is no magnetic force on the loop
- Outward along the radius of the loop
- Upward along the axis of the loop
- Downward along the axis of the loop
- Inward along the radius of the loop

### Solution 3

The Right Hand Rule

The magnetic field of I_{1} would be tangential to the imaginary circle traced by the fingers. In this case, *l* and *B* are in the same direction

There is no magnetic force on the loop. The answer is **A**.

### Problem 4

A very long thin straight wire carries a uniform charge density of λ per unit length. WHich of the following gives the manitude of the electric field at a radial distance r from the wire?

- \frac{1}{4 \pi \epsilon_0} \lambda \mathrm{ln} r
- \frac{1}{2 \pi \epsilon_0} \frac{r}{\lambda}
- \frac{1}{2 \pi \epsilon_0} \frac{\lambda}{r^2}
- \frac{1}{4 \pi \epsilon_0} \frac{\lambda^2}{r^2}
- \frac{1}{2 \pi \epsilon_0} \frac{\lambda}{r}

### Solution 4

E### Problem 5

The bar magnet shown in the figure above is moved completely through the loop. Which of the following is a true statement abotu the direction of the current flow between the two points a and b in the circuit?

- Current flows from b to a as the magnet enters the loop and from a to b as the magnet leaves the loop
- Current flows from a to b as the magnet passes through the loop
- No current flows between a and b as the magnet passes through the loop
- Current flows from b to a as the magnet passes through the loop
- Current flows from a to b as the magnet enters the loop and from b to a as the magnet leaves the loop

### Solution 5

A### Problem 6

Consider 2 very long straight insulated wires oriented at right angles. The wires carry currents of equal manitude I in the directions shown in the figure aboe. What is the net magnetic field at point P?

### Solution 6

### Problem 7

3 long straight wires in the xz-plane each carrying current I, cross at the origin of coordinates.

### Solution 7

### Problem 8

A particle with mass m and charge q, moving with a velocity v, enters a region of uniform magnetic field B. The particle strikes the wall at a distance d from the entrance slit. If the particle's velocity stays the same but its charge-to-mass ratio is doubled, at what distance from the entrance slit will the particle strike the wall?

### Solution 8

### Problem 9

Consider the closed cylindrical Gaussian surface above. Suppose that the net charge enclosed within this surface is +1E-9C and the electric flux out throug the portion of the surface marked A is -100 Nm^{2}/C. The flux through the rest of the surface is most nearly how many Nm^{2}/C?

### Solution 9

### Problem 10

A large, parallel-plate capacitor consists of two square plates that measure 0.5 m on each side. A charging current of 9 A is applied to the capacitor. How many V/ms is the approximate rate of change of the electric field between the plates?

### Solution 10

### Problem 11

A wire loop that encloses an area of 10 cm^{2} has a resistance of 5 Ω. The loop is placed in a magnetic field of 0.5 T with its plane perpendicular to the field. The loop is suddenly removed from the field. How much charge flows past a given point in the wire?

### Solution 11

### Problem 12

### Solution 12

### Problem 13

Two thin, concentric, spherical conducting shells are arranged as shown in the figure above. The inner shell has radius a, charge +Q, and is at zero electric potential. The outer shell has radius b and charge -Q. If r is the radial distance from the center of the spheres, what is the electric potential in region I (a < r < b) and in region II (r > b)?

### Solution 13

### Problem 14

In static electromagnetism, let **E**, **B**, **J**, and ρ be the electric field, magnetic field, current density, and charge density, respectively. Which of the following conditions allows the electric fiedlt o be written in the form

where φ is the electrostatic potential?

### Solution 14

### Problem 15

A long, straight, hollow cylindrical wire with an inner radius R and and outer radius 2R carries a uniform current density. Which of the following graphs best represents the magnitude of the magnetic field as a function of the distance from the center of the wire?

### Solution 15

E

### Problem 16

A parallel-plate capacitor has plate deparation d. THe space between the plats is empty. A battery supplying voltage V_{0} is connected acorss the capacitor, resulting in electromagnetic energy U_{0} stored in the capacitor. A dielectric, of dielectric constant κ, is inserted so that it just fills the space between the plates. If the battery is still connected, what are the electric field E and the energy U stored in the dielecctric, in terms of V_{0} and U_{0}

- E = \frac{V_0}{d} , U = \kappa^2 U_0
- E = \frac{V_0}{\kappa d} , U = \kappa U_0
- E = \frac{V_0}{d} , U = U_0
- E = \frac{V_0}{d} , U = \kappa^2 U_0
- E = \frac{V_0}{\kappa d} , U = U_0

### Solution 16

D

### Problem 17

One end of a Nichrome wire of length 2L and cross-sectional area A is attached to an end of another Nichrome wire of length L and crosssectional area 2A. If the free end of the longer wire is at an electric potential of 8.0 volts, and the free end of the shorter wire is at an electric potential of 1.0 volt, what is the approximate potential at the junction of the two wires?

### Solution 17

Electric Potential

Recall the equation for resistance

*L* is the length of the wire (in meters). *A* is the cross-sectional area of the wire (in meters^{2}). *ρ* is the resistivity of the material (in ohm meter)

The long and thin wire's resistance is

The fat and short wire's resistance is

The resistors are in a series, so the equivalent resistance is

Use Ohm's Law to find the current *I*

To find the potential at the junction, use Ohm's Law again to get

The 1 comes from the fact that the potential at the end of the fatter wire is 1V, and thus the voltage across the junction is just the voltage at the end of the fat wire plus the current times the resistance of the fatter wire.

### Problem 18

For blue light, a transparent material has a relative permittivity (dielectric constant) of 2.1 and a relative permeability of 1.0. If the speed of light in a vacuum is c, what is the phase velocity of blue light in an unbounded medium of this material?

### Solution 18

So everyone knows that the speed of light is 3.00E8 m/s right?

Not necessarily!

It depends on the medium. The speed of light is actually a function of permittivityε and permability μ. The equation is:

### Problem 19

A wire of diameter 0.02m contains 1E28 free electrons per cubic meter. For an electric current of 100 amperes, what is the drift velocity for free electrons in the wire?

### Solution 19

The given information is diameter d = 0.02m, so radius is 0.01m. Current density J = 1E28 electrons/meter. Current I = 100A. The charge of an electron is e = 1.6E-19 Coulombs.

Use this equation

where the area of a circle A = π r^{2}. Then use this equation

where n = N / V, the number of electrons per volume was given to us in the question. e is known. We're trying to find v. So from the above two equations, get this equation

Solve for A

Plug A in to v

### Problem 20

An isolated sphere of radius R contains a uniform volumn distribution of positive charge. Which of the curves (A through E) on the graph above correctly illustrates the dependence of the magnitude of the electric field of the sphere as a function of the distance r from the center?

### Solution 20

C

### Problem 21

Which of the following statements most accurately describes how an electromagnetic field behaves under a Lorentz transformation?

- The electric field transforms completely into a magnetic field
- If initially there is only an electric field, after the transformation there may be both an electric and a magnetic field
- The electric field is unaltered
- The magnetic field is unaltered
- It cannot be determined unless a gauge transformation is also specified

### Solution 21

If initially there is only an electric field, after the transformation there may be both an electric and a magnetic field

### Problem 22

Which of the following statements concerning the electric conductivities at room temperature of a pure copper sample and a pure silicon sample is FALSE?

- If the temperature of the copper sample is increased, its conductivity will decrease.
- The addition of an impurity in the copper sample always decreases its conductivity
- The addition of an impurity in the copper sample always decreases its conductivity
- The conductivity of the copper sample is many orders of magnitude greater than that of the silicon sample
- If the temperature of the silicon sample is increase, its conductivity will increase

### Solution 22

The addition of an impurity in the copper sample always decreases its conductivity

### Problem 23

The battery in the diagram above will be charged by the generator G. The generator has a terminal voltage of 120 volts when the charging current is 10 amperes. The battery has an emf of 100 volts and an internal resistance of 1 ohm. In order to charge the battery at 10 amperes charging current, how many ohms should we set thte resistance R?

### Solution 23

1 ohm

### Problem 24

A negative test charge is moving near a long straight wire which there is a current. A force will act on the test charge in a direction parallel to the direction of the current if the motion of the charge is in a direction

- away from the wire
- opposite to that of the current
- toward the wire
- the same as that of the current
- perpendictular to both the direction of the current and the direction toward the wire

### Solution 24

toward the wire

### Problem 25

A charged particle oscillates harmonically along the x-maxis as shown above. The radiation from the particle is detected at a distance point P, which lies in the xy-plane. The electric field at P is in **which** direction/plane and has a maximum aplitude at θ = **what**?

### Solution 25

xy plane, θ = 90 degrees

### Problem 26

A dielectric of dielectric constant K is placed in contact with a conductor having surface charge density σ as shown above. What is the polarization (bound) charge density σ_{p} on the surface of the dielectric at the interface between the two materials?

### Solution 26

### Problem 27

One of Maxwell's equations

Which of the following sketches shows magnetic field lines that clearly violate this equation within the region bounded by the dashed lines?

### Solution 27

D

### Problem 28

A small circular wire loop of radius a is located at the center of a much larger circular wire loop of radius b, as shown above. The large loop carries an alternating current I = I_{0} cos ω t, where I_{0} and ω are constants. The magnetic field generated by the curren in the large loop induces in the small lop an emf that is approximately equal to which of the following? (Either use mks units and like μ_{0} be the permeability of free space, or use Gaussian units and let μ_{0} be 4 π / c^{2}

### Solution 28

### Problem 29

The long thin cylindrical glass rod shown above has length ℓ and is insulted from its surroundings. The rod has an excess charge Q uniformly distributed along its length. Assume the electric potential to be zero at infinite distances from the rod. If k is the constant in Couloumbs law, the electric potential at a point P along the axis of the rod and a distance ℓ from one end is k Q / ℓ multiplied by what?

### Solution 29

### Problem 30

A coaxial cable having radii a, b, and c carries equal and opposite currents of magnitude i on the inner and outer conductors. What is the magnitude of the magnetic induction at point P outside of the cable at a distance r from the axis?

- Zero
- \frac{\mu_0 i r}{2\pi a^2}
- \frac{\mu_0 i r}{2\pi a^2}
- \frac{\mu_0 i r}{2\pi a^2}
- \frac{\mu_0 i r}{2\pi a^2}

### Solution 30

Magnetic induction is magnetic field. The opposite currents cancel each other. Therefore, magnetic field outside is 0. The answer is A

### Problem 31

Two large conducting plates form a wedge of angle α as shown in the diagram above. The plates are insulated from each other; one has a potential V_{0} and the other is grounded. Assuming that the plates are large enough so that the potential difference between them is independenet of the cylindrical coordinates z and ρ, the potential anywhere between the plates as a function of the angle φ is

*V*_{0}/*α**V*_{0}*α*/*φ**V*_{0}*φ*^{2}/*α**V*_{0}*φ*/*α**V*_{0}*α*/*φ*^{2}

### Solution 31

A Potential V is related to the electric field E by

Since the problem supplies the approximation tool that the planes are quite large, one can assume the field is approximately constant. The remaining parameter that can't be thrown out by this approximation is the angle, and thus the only choice that yields

is choice A.

### Problem 32

If magnetic monopoles exist, which of these Maxwell's equations of electromagnetism would be INCORRECT?

- \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}
- \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
- \nabla \cdot \mathbf{D} = \rho
- \nabla \cdot \mathbf{B} = 0

### Solution 32

Magnetic monopoles remain a likeable (even lovable) theoretical construct because of their ability to perfectly symmetrize Maxwell's equations. Since the curl term has an electric current, the other curl term should have a magnetic current.

is taken to be obvious in presence of magnetic charge. The answer is II and IV

The revised equations are:

- \nabla \times \mathbf{H} = \mathbf{J}_e + \frac{\partial \mathbf{D}}{\partial t}
- \nabla \times \mathbf{E} = -\mathbf{J}_m - \frac{\partial \mathbf{B}}{\partial t}
- \nabla \cdot \mathbf{D} = \rho_e
- \nabla \cdot \mathbf{B} = \rho_m

### Problem 33

A conducting cavity is driven as an electromagnetic resonator. If perfect conductivity is assumed, the transverse and normal field components must obey which of the following conditions at the inner cavity walls?

*E*= 0_{n}*E*= 0_{t}*B*= 0_{n}*B*= 0_{t}

### Solution 33

The full formalism of a conducting cavity can be solved via TEM (transverse electromagnetic) wave guides. However, to solve this problem, one needs only the two boundary conditions from the reflection at a conducting surface,

and

The electric field parallel to the cavity is the transverse field. E is a curl-less field (i.e. it only diverges), so for the E-field lines to be parallel with the inner walls of an enclosure is just silly. B is a divergence-less field, (i.e. it only curls), so for the B-field lines to be perpendicular with the inner walls of the enclosure is equally as silly.

*E _{t}* = 0 and

*B*= 0

_{n}### Problem 34

A plane-polarized electromagnetic wave is incident normally on a flat, perfectly conducting surface. Upon reflection at the surface, which of the following is true?

- The magnetic vector is reversed; the electric vector is not.
- The electric vector is reversed; the magnetic vector is not.
- The directions of the electric and magnetic vectors are interchanged.
- Neither the electric vector nor the magnetic vector is reversed.
- Both the electric vector and magnetic vector are reversed.

### Solution 34

The answer is B. The electric field is reversed. The magnetic field doubles but the direction stays the same.

### Problem 35

A rectangular loop of wire with dimensions shown above is coplanar with a long wire carrying current I. The distance between the wire and the left side of the loop is r. The loop is pulled to the right as indicated.

What are the directions of the induced current in the loop and the magnetic forces on the left and the right sides of the loop as the loop is pulled?

Induced Current ... Force on Left Side ... Force on Right Side

### Solution 35

Clockwise ... To the left ... To the right

### Problem 36

What is the magnitude of the net force on the loop when the induced current is i?

### Solution 36

### Problem 37

A uniform and constant magnetic field **B** is directed perpendicularly into the plane of the page everywhere within a rectangular region as shown above. A wire circuit in the shape of a semicircle is uniformly rotated counterclockwise in the plane of the page about an axis A. The axis A is perpendicular to the page at the edge of the field and directed through the center of the straight-line portion of the circuit. Which of the following graphics best approximates the emf ε induced in the circuit as a function of t?

### Solution 37

Faraday's Law has the induced voltage is given by the change in magnetic flux

(The minus sign shows that the induced voltage opposes the change.)

Since the induced voltage has to be periodic (as the half-circle rotates around A), choices (D) and (E) are immediately eliminated.

The voltage changes from positive to negative in regions where the change in flux is slowing down, goes to 0, then speeds up again. Thus, choice (C) is out.

The change in flux is constantly increasing as the loop spins into the field, and it is constantly decreasing as it spins out of the field. This is choice (A).

### Problem 38

If an electric field is given in a certain region by E_{x}=0,E_{y}=0,E_{z}=kz, where k is a nonzero constant, which of the following is true?

- There is a time-varying magnetic field.
- The electric field is impossible under any circumstances.
- The electric field cannot be constant in time.
- There is charge density in the region.
- None of these choices are correct

### Solution 38

Gauss Law says the divergence of E in Cartesian coordinates is non-zero

The answer is D

### Problem 39

In the RLC circuit shown above, the applied voltage is

For a constant ε_{m}, at what angular frequency ω does the current have its maximum steady-state amplitude after the transients have died out?

### Solution 39

The maximum steady-state amplitude (after transients die out) occurs at the resonant frequency, which is given by setting the impedance of the capacitor and inductor

### Problem 40

Two pith balls of equal mass M and equal charge q are suspended from the same point on long massless threads of length L as shown in the figure above. If k is the Coulomb's law constant, then for small values of θ, what is the distance d between the charged pith balls at equilibrium?

### Solution 40

### Problem 41

An electron oscillates back and forth along the + and -x axes, consequently emitting electromagnetic radiation. Which of the following statements concerning the radiation is FALSE?

- Far from the electron, the rate at which radiated energy crosses a perpendicular unit area is a maximum when the unit area is located on the + or - x-axes.
- Far from the electron, the rate at which radiated energy crosses a perpendicular unit area decreases as the inverse square of the distance from the electron.
- The total rate of radiation of energy into all directions is proportional to the square of the electron's charge.
- The total rate of radiation of energy into all directions is proportional to the square of the electron's acceleration.
- Far from the electron, the radiated energy is carried equally by the transverse electric and the transverse magnetic fields.

### Solution 41

A

### Problem 42

The circuit shown above is used to measure the size of the capacitance C. The y-coordinate of the spot on the oscilloscope screen is propotional to the potential difference acress R, and the x-coordinate of the spot is swept at a constant speed s. The switch is closed and then opened. One can then calculate C from the shape and the size of the curve on the screen plus a knowledge of which of the following?

- R and R'
- s and V
_{0} - s and R
- V
_{0}and R - The sensitivity of the oscilloscope

### Solution 42

C

### Problem 43

A parallel-plate capacitor is connected to a battery. V_{0} is the potential difference between the plates, Q_{0} is the charge on the positive plate, E_{0} the magnitude of the electric field, and D_{0} the magnitude of the displacement vector. The original vacuum between the plates is filled with a dielectric and then the battery is disconnected. If the corresponding electrical parameters for the final state of the capacitor are deonted by a subscript f, which of the following is true?

*D*>_{f}*D*_{0}*V*<_{f}*V*_{0}*E*>_{f}*E*_{0}*Q*=_{f}*Q*_{0}*V*>_{f}*V*_{0}

### Solution 43

*D _{f}* >

*D*

_{0}

### Problem 44

A flat coil of wire is rotated at a frequency of 10 hertz in the magnetic field produced by three pairs of magnets as shown above. The axis of rotation of the coil lies in the plane of the coil and is perpendicular to the field lines. What is the frequency of the alternating voltage in the coil?

### Solution 44

A three-pole magnet should produce three voltage peaks. The frequency is 30 Hz.

### Problem 45

The circuit shown above is in a uniform magnetic field that is into the page and is decreasing in magnitude at the rate of 150 tesla/second. The ammeter reads:

- 0.15 A
- 0.35 A
- 0.50 A
- 0.65 A
- 0.80 A

### Solution 45

Answer

### Problem 46

The electric potential at a point P which is located on the axis of symmetry a distance x from the center of the ring, is given by:

### Solution 46

Answer

### Problem 47

A small particle of mass m and charge -q is placed at point P and released. If R >> x, the particle will undergo oscillations along the axis of symmetry with an angular frequency that is equal to:

### Solution 47

Answer

### Problem 48

The circuits below consist of 2-element combinations of capacitors, diodes, and resistors. V_{in} represents an ac-voltage with variable frequency. it is desired to build a circuit for which V_{out} is approximately equal to V_{in} at high frequencies and V_{out} is approximately equal to 0 at low frequencies. Which of the following circuits will preform this task?

### Solution 48

Answer

### Problem 49

A circular wire loop of radius R rotates with an angular speed ω in a uniform magnetic field **B**, as shown in the figure above. If the emf ε induced in the loop is

then what is the angular speed of the loop?

### Solution 49

Answer

### Problem 50

A wire is being wound around a rotating wooden cylinder of radius R. One end of the wire is connected to the axis of the cylinder as shown in the figure above. The cylinder is placed in a uniform magnetic field of magnitude B parallel to its axis and rotates at N revolutions per second. What is the potential difference between the open ends of the wire?

### Solution 50

Answer

### Problem 51

Two long conductors are arranged as shown above to form overlapping cylinders, each of radius r, whose centers are separated by a distance d. Current of density J flows into the plane of the page along the shaded part of one conductor and an equal current flows out of the plane of the page along the shaded portion of the other. What are the magnitude and direction fo the magnetic field at point A?

### Solution 51

Answer

### Problem 52

The figure above shows a trajectory of a particle that is deflecte as it moves through the uniform electric field between parallel plates. There is potential difference V and distance d between the plates and they have length L. The particle (mass m, charge q) has nonrelativistic speed v before it enters the field, and its direction at this time is perpendicular to the field. For small deflections, which of the following expressions is the best approximation to the deflection angle θ?

### Solution 52

Answer

### Problem 53

A positively charged particle is moving in the xy-plane in a region wwhere there is non-zero uniform magnetic field B in the +z direction and a non-zero uniform electric field E in the +y direction. Which of the following is a possible trajectory for the particle?

### Solution 53

Answer

### Problem 54

Two small pith balls, each carrying a charge q, are attached to the ends of a light rod of length d, which is suspended from the ceiling by a thin torsion-free fiber. There is a uniform magnetic field B, point straight down, in the cylindrical region of radius R around the fiber. The system is initially at rest. if the magnetic field is turned off, which of the following describes what happens to the system?

- It rotates with angular momentum (1/2)qBRd
- It does not move because magnetic forces do no work
- It rotates with angular momentum qBR
^{2} - It does not rotate because to do so would violate conservation of angular momentum
- It rotates with angular momentum (1/4)qBd
^{2}

### Solution 54

C

### Problem 55

The coaxial cable has the cross section shown in the figure above. The shaded region is insulated. The regions in which r < a and b < r < c are conducting. A uniform dc current density of total current I flows along the inner part of the cable ( r < a ) and returns along the outer part of the able (b < r < c) in the directions shown. The radial dependence of the magnitude of the magnetic field, H, is shown by which of the following?

### Solution 55

Answer

### Problem 56

A particle with charge q and momentum p is moving in the horizontal plane under the action of a unform vertical magnetic field of magnitude B. Measurements are made of the particle's trajectory to determind the sagitta s and half-chord length ℓ. Which of the following expressions gives the particle's momentum in terms of q, B, s, and ℓ? (assume s << ℓ).

### Solution 56

Answer