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Electromagnetism

What is an Electric Field?

E = \frac{F}{q_0}

Coulomb's Law

F = \frac{Q_1 Q_2}{4 \pi \epsilon_0 d^2}

Gauss's Law

\mathbf{\psi_{net}} = \oint_s D \cdot d s = Q_s

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

  1. R = ρ π b2 / l
  2. R = ρ l / π b2
  3. R = ρ l / π (b2 - a2)
  4. R = ρ l / π a2
  5. R = ρ π (b2 - a2) / l

Solution 1

I knew the answer would include b2 - a2, so C and E would be good guesses. To start, get the current density:

j = \frac{I}{A} = \frac{I}{\rho (b^2 - a^2)}
By Ohm's Law
j = \sigma E = \frac{1}{\rho}= \frac{1}{\rho} E = \frac{1}{\rho}\frac{V}{l}
Where s is the conductivity and r is s the resistivity
\frac{1}{\rho}\frac{V}{l} = \frac{1}{\pi(b^2 - a^2)}
Finally
R = \frac{V}{I} = \frac{\rho l}{\pi (b^2 - a^2)}

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

504

An infinitely long, straight wire carrying current I1 passes through the center of a circular loop of wire carrying current I2. The long wire is perpendicular to the plane of the loop. Which of the following describes the magnetic force on the loop?

  1. Inward along the radius of the loop
  2. Upward along the axis of the loop
  3. Downward along the axis of the loop
  4. Outward along the radius of the loop
  5. There is no magnetic force on the loop

Solution 3

\vec{F} = I \vec{l} \times \vec{B} = 0

The Right Hand Rule

504

The magnetic field of I1 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 E.

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?

  1. \frac{1}{4 \pi \epsilon_0} \lambda \mathrm{ln} r
  2. \frac{1}{4 \pi \epsilon_0} \frac{\lambda^2}{r^2}
  3. \frac{1}{2 \pi \epsilon_0} \frac{r}{\lambda}
  4. \frac{1}{2 \pi \epsilon_0} \frac{\lambda}{r}
  5. \frac{1}{2 \pi \epsilon_0} \frac{\lambda}{r^2}

Solution 4

D

Problem 5

518

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?

  1. Current flows from b to a as the magnet enters the loop and from a to b as the magnet leaves the loop
  2. Current flows from a to b as the magnet enters the loop and from b to a as the magnet leaves the loop
  3. No current flows between a and b as the magnet passes through the loop
  4. Current flows from a to b as the magnet passes through the loop
  5. Current flows from b to a as the magnet passes through the loop

Solution 5

A

Problem 6

538

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

560

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

Solution 7

Problem 8

561

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

562

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 Nm2/C. The flux through the rest of the surface is most nearly how many Nm2/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 cm2 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

571

Solution 12

Problem 13

590

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?

  1. 592
  2. 592
  3. 592
  4. 592
  5. 592

Solution 15

C

Problem 16

A parallel-plate capacitor has plate deparation d. THe space between the plats is empty. A battery supplying voltage V0 is connected acorss the capacitor, resulting in electromagnetic energy U0 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 V0 and U0

  1. E = \frac{V_0}{\kappa d} , U = U_0
  2. E = \frac{V_0}{\kappa d} , U = \kappa U_0
  3. E = \frac{V_0}{d} , U = \kappa^2 U_0
  4. E = \frac{V_0}{d} , U = \kappa^2 U_0
  5. E = \frac{V_0}{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

\Delta V = 8 - 1 = 7 \mathrm{volts}

Recall the equation for resistance

R = \rho \frac{L}{A}

L is the length of the wire (in meters). A is the cross-sectional area of the wire (in meters2). ρ is the resistivity of the material (in ohm meter)

The long and thin wire's resistance is

R_1 = \rho \frac{2L}{A}

The fat and short wire's resistance is

R_2 = \rho \frac{L}{2A}

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

R_{\mathrm{eqiv}} = (2+ \frac{1}{2}) \rho \frac{L}{A}
R_{\mathrm{eqiv}} = (\frac{4}{2} + \frac{1}{2}) \rho \frac{L}{A}
R_{\mathrm{eqiv}} = \frac{5 \rho L}{2 A}

Use Ohm's Law to find the current I

V = I R
7 = I \frac{5 \rho L}{2 A}
I = \frac{14 A}{5 \rho L}

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

V_{\mathrm{junc}} = 1 + I R_2

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.

V_{\mathrm{junc}} = 1 + (\frac{14 A}{5 \rho L}) (\frac{\rho L}{2A})=
V_{\mathrm{junc}} = \frac{5}{5} + \frac{7}{5} = \frac{12}{5}
V_{\mathrm{junc}} = 2.4 \mathrm{volts}

Problem 18

703

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:

\frac{1}{\sqrt{\mu \epsilon }}
where
c = \frac{1}{\sqrt{\mu_0 \epsilon_0 }}
So plug the given information into the first question
v = \frac{1}{\sqrt{( 1.0 \mu_0 ) ( 2.1 \epsilon_0 ) }}
=\frac{1}{\sqrt{ \mu_0 \epsilon_0 }} \frac{1}{\sqrt{( 1.0 ) ( 2.1 ) }}
= (c) \frac{1}{\sqrt{2.1}}
=\frac{c}{\sqrt{2.1}}

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

J = \frac{I}{A}

where the area of a circle A = π r2. Then use this equation

J = n e v

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

\frac{I}{A} = n e v
v =\frac{I}{A n e}

Solve for A

A = \pi r^2
= \pi (\frac{d}{2})^2
= \pi (\frac{0.02}{2})^2 = \pi (0.01)^2
= \pi (1E-4) \mathrm \approx 3.14 \times 10^{-4}

Plug A in to v

v = \frac{100}{ (3.14 \times 10^{-4}) (1 \times 10^{28}) (1.60 \times 10^{-19})}
= \frac{1 \times 10^2}{ (3.14 \times 10^{24}) (1.60 \times 10^{-19}) }
= \frac{1 \times 10^2}{ 5.02 \times 10^5 }
= 0.20 \times 10^{-3}
= 2 \times 10^{-4}

Problem 20

710

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?

  1. The electric field is unaltered
  2. If initially there is only an electric field, after the transformation there may be both an electric and a magnetic field
  3. The electric field transforms completely into a magnetic field
  4. The magnetic field is unaltered
  5. 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?

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

Solution 22

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

Problem 23

724

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

  1. away from the wire
  2. perpendictular to both the direction of the current and the direction toward the wire
  3. toward the wire
  4. opposite to that of the current
  5. the same as that of the current

Solution 24

toward the wire

Problem 25

753

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

754

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 σp on the surface of the dielectric at the interface between the two materials?

Solution 26

\sigma \frac{1-K}{K}

Problem 27

One of Maxwell's equations

\nabla \cdot \mathrm{B} = 0

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

779

Solution 27

D

Problem 28

781

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 = I0 cos ω t, where I0 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 μ0 be the permeability of free space, or use Gaussian units and let μ0 be 4 π / c2

Solution 28

Problem 29

798

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

809

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?

  1. \frac{\mu_0 i r}{2\pi a^2}
  2. \frac{\mu_0 i r}{2\pi a^2}
  3. \frac{\mu_0 i r}{2\pi a^2}
  4. \frac{\mu_0 i r}{2\pi a^2}
  5. Zero

Solution 30

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

Problem 31

812

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 V0 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

  1. V0 φ / α
  2. V0 α / φ
  3. V0 α / φ2
  4. V0 / α
  5. V0 φ2 / α

Solution 31

A Potential V is related to the electric field E by

\vec{E}=\nabla V

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

\frac{d}{d\phi} V = \mathrm{constant}

is choice D.

Problem 32

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

  1. \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}
  2. \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
  3. \nabla \cdot \mathbf{D} = \rho
  4. \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.

\nabla \cdot B \neq 0

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

The revised equations are:

  1. \nabla \times \mathbf{H} = \mathbf{J}_e + \frac{\partial \mathbf{D}}{\partial t}
  2. \nabla \times \mathbf{E} = -\mathbf{J}_m - \frac{\partial \mathbf{B}}{\partial t}
  3. \nabla \cdot \mathbf{D} = \rho_e
  4. \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?

  1. En = 0
  2. Et = 0
  3. Bn = 0
  4. Bt = 0

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,

\Delta E_{\parallel} = 0

and

\Delta B_{\perp} = 0

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.

Et = 0 and Bn = 0

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?

  1. Neither the electric vector nor the magnetic vector is reversed.
  2. The directions of the electric and magnetic vectors are interchanged.
  3. The electric vector is reversed; the magnetic vector is not.
  4. Both the electric vector and magnetic vector are reversed.
  5. The magnetic vector is reversed; the electric vector is not.

Solution 34

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

Problem 35

854

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

855

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

Solution 36

\frac{\mu_0 i I}{2 \pi} \mathrm{ln}(\frac{a b}{r ( r + a )})

Problem 37

857

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?

857

Solution 37

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

V = -\frac{dB \cdot A}{dt}

(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 Ex=0,Ey=0,Ez=kz, where k is a nonzero constant, which of the following is true?

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

Solution 38

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

\nabla \cdot \vec{E} = \rho / \epsilon_0

The answer is C

Problem 39

881

In the RLC circuit shown above, the applied voltage is

\epsilon (t) = \epsilon_m \mathrm{cos} \omega t

For a constant εm, 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

X_C = X_L
\frac{1}{\omega C} = \omega L
\omega^2 =1/(LC)
\omega = \frac{1}{\sqrt{LC}}

Problem 40

883

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

(\frac{2 k q^2 L}{M g})^{1/3}

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?

  1. 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.
  2. Far from the electron, the radiated energy is carried equally by the transverse electric and the transverse magnetic fields.
  3. The total rate of radiation of energy into all directions is proportional to the square of the electron's acceleration.
  4. The total rate of radiation of energy into all directions is proportional to the square of the electron's charge.
  5. 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.

Solution 41

A

Problem 42

886

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?

  1. R and R'
  2. s and V0
  3. The sensitivity of the oscilloscope
  4. s and R
  5. V0 and R

Solution 42

D

Problem 43

A parallel-plate capacitor is connected to a battery. V0 is the potential difference between the plates, Q0 is the charge on the positive plate, E0 the magnitude of the electric field, and D0 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?

  1. Df > D0
  2. Ef > E0
  3. Vf < V0
  4. Vf > V0
  5. Qf = Q0

Solution 43

Df > D0

Problem 44

892

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

902

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:

  1. 0.15 A
  2. 0.35 A
  3. 0.50 A
  4. 0.65 A
  5. 0.80 A

Solution 45

Answer

Problem 46

903
This is a thin nonconducting ring of radius R which has a charge Q uniformly spread out on it

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

904
This is a thin nonconducting ring of radius R which has a charge Q uniformly spread out on it

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. Vin represents an ac-voltage with variable frequency. it is desired to build a circuit for which Vout is approximately equal to Vin at high frequencies and Vout is approximately equal to 0 at low frequencies. Which of the following circuits will preform this task?

945

Solution 48

Answer

Problem 49

946

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

\epsilon_0 \mathrm{sin} \omega t

then what is the angular speed of the loop?

Solution 49

Answer

Problem 50

947

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

969

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

971

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?

986

Solution 53

Answer

Problem 54

987

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?

  1. It does not rotate because to do so would violate conservation of angular momentum
  2. It rotates with angular momentum qBR2
  3. It rotates with angular momentum (1/4)qBd2
  4. It rotates with angular momentum (1/2)qBRd
  5. It does not move because magnetic forces do no work

Solution 54

B

Problem 55

988

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?

988

Solution 55

Answer

Problem 56

989

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