Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

In an electrical storm, the cloud and the ground act like a parallel-plate capacitor, which typically charges up due to frictional electricity in collisions of ice particles in the cold upper atmosphere. Lightning occurs when the magnitude of the electric field builds up to a critical value, E _{c}, at which air is ionized.

(a)
Treat the cloud as a flat square with sides of length L. If it is at a height h above the ground, find the amount of energy released in the lightning strike.

(b)
Based on your answer from part (a), which is more dangerous, a lightning strike from a high-altitude cloud or a low-altitude one?

(c)
Make an order-of-magnitude estimate of the energy released by a typical lightning bolt, assuming reasonable values for its size and altitude. E _{c }is about 10^{6} V/m.

See problem 21 for a note on how recent research affects this estimate.

2

The neuron in the figure has been drawn fairly short, but some neurons in your spinal cord have tails (axons) up to a meter long. The inner and outer surfaces of the membrane act as the "plates" of a capacitor. (The fact that it has been rolled up into a cylinder has very little effect.) In order to function, the neuron must create a voltage difference V between the inner and outer surfaces of the membrane. Let the membrane's thickness, radius, and length be t, r, and L.

(a)
Calculate the energy that must be stored in the electric field for the neuron to do its job, simplifying your result as much as possible. (In real life, the membrane is made out of a substance called a dielectric, whose electrical properties increase the amount of energy that must be stored. For the sake of this analysis, ignore this fact.) [Hint: The volume of the membrane is essentially the same as if it was unrolled and flattened out.]

(b)
An organism's evolutionary fitness should be better if it needs less energy to operate its nervous system. Based on your answer to part (a), what would you expect evolution to do to the dimensions t and r? Why don't similar evolutionary pressures apply to L? What other constraints would keep these evolutionary trends from going too far?

3

Consider two solenoids, one of which is smaller so that it can be put inside the other. Assume they are long enough so that each one only contributes significantly to the field inside itself, and the interior fields are nearly uniform. Consider the configuration where the small one is inside the big one with their currents circulating in the same direction, and a second configuration in which the currents circulate in opposite directions. Compare the energies of these configurations with the energy when the solenoids are far apart. Based on this reasoning, which configuration is stable, and in which configuration will the little solenoid tend to get twisted around or spit out? [Hint: A stable system has low energy; energy would have to be added to change its configuration.]

4

The figure shows a nested pair of circular wire loops used to create
magnetic fields. (The twisting of the leads is a practical trick for reducing
the magnetic fields they contribute, so the fields are very nearly what we
would expect for an ideal circular current loop.) The coordinate system
below is to make it easier to discuss directions in space. One loop is in the
y-z plane, the other in the x-y plane. Each of the loops has a radius of
1.0 cm, and carries 1.0 A in the direction indicated by the arrow.

(a) Using the equation in optional section 6.2, calculate the magnetic
field that would be produced by one such loop, at its center.

(b)
Describe the direction of the magnetic field that would be produced,
at its center, by the loop in the x-y plane alone.

(c)
Do the same for the other loop.

(d) Calculate the magnitude of the magnetic field produced by the two
loops in combination, at their common center. Describe its direction.

√

5

(a) Show that the quantity

has units of velocity.

(b)
Calculate it numerically and show that it equals the speed of light.

(c)
Prove that in an electromagnetic wave, half the energy is in the electric
field and half in the magnetic field.

6

One model of the hydrogen atom has the electron circling around the
proton at a speed of 2.2x10^{6} m/s, in an orbit with a radius of 0.05 nm. In
homework problem 9 in chapter 3, you calculated the current created by
the electron.

(a) Now estimate the magnetic field created at the center of the atom by
the electron. We are treating the circling electron as a current loop, even
though it's only a single particle.

(b)
Does the proton experience a nonzero force from the electron's
magnetic field? Explain.

(c)
Does the electron experience a magnetic field from the proton? Ex
plain.

(d)
Does the electron experience a magnetic field created by its own
current? Explain.

(e) Is there an electric force acting between the proton and electron? If
so, calculate it.

(f)
Is there a gravitational force acting between the proton and electron? If
so, calculate it.

(g)
An inward force is required to keep the electron in its orbit - other
wise it would obey Newton's first law and go straight, leaving the atom.
Based on your answers to the previous parts, which force or forces (elec
tric, magnetic and gravitational) contributes significantly to this inward
force?
[Based on a problem by Arnold Arons.]

a) + e) √

7

[You need to have read optional section 6.2 to do this problem.] Suppose a charged particle is moving through a region of space in which there is an electric field perpendicular to its velocity vector, and also a magnetic field perpendicular to both the particle's velocity vector and the electric field. Show that there will be one particular velocity at which the particle can be moving that results in a total force of zero on it. Relate this velocity to the magnitudes of the electric and magnetic fields. (Such an arrangement, called a velocity filter, is one way of determining the speed of an unknown particle.)

8

If you put four times more current through a solenoid, how many times more energy is stored in its magnetic field?

9

Suppose we are given a permanent magnet with a complicated, asymmetric shape. Describe how a series of measurements with a magnetic compass could be used to determine the strength and direction of its magnetic field at some point of interest. Assume that you are only able to see the direction to which the compass needle settles; you cannot measure the torque acting on it.

*

10

Consider two solenoids, one of which is smaller so that it can be put inside the other. Assume they are long enough to act like ideal solenoids, so that each one only contributes significantly to the field inside itself, and the interior fields are nearly uniform. Consider the configuration where the small one is partly inside and partly hanging out of the big one, with their currents circulating in the same direction. Their axes are constrained to coincide.

(a)
Find the magnetic potential energy as a function of the length x of the part of the small solenoid that is inside the big one. (Your equation will include other relevant variables describing the two solenoids.)

(b)
Based on your answer to part (a), find the force acting between the solenoids.

11

Four long wires are arranged, as shown, so that their cross-section forms a square, with connections at the ends so that current flows through all four before exiting.

Note that the current is to the right in the two back wires, but to the left in the front wires. If the dimensions of the cross-sectional square (height and front-to-back) are b, find the magnetic field (magnitude and direction) along the long central axis.

12

To do this problem, you need to understand how to do volume integrals in cylindrical and spherical coordinates. (a) Show that if you try to integrate the energy stored in the field of a long, straight wire, the resulting energy per unit length diverges both at r→0 and r→∞. Taken at face value, this would imply that a certain reallife process, the initiation of a current in a wire, would be impossible, because it would require changing from a state of zero magnetic energy to a state of infinite magnetic energy. (b) Explain why the infinities at r→0 and r→∞ don't really happen in a realistic situation. (c) Show that the electric energy of a point charge diverges at r→0, but not at r→∞.

A remark regarding part (c): Nature does seem to supply us with particles that are charged and pointlike, e.g. the electron, but one could argue that the infinite energy is not really a problem, because an electron traveling around and doing things neither gains nor loses infinite energy; only an infinite change in potential energy would be physically troublesome. However, there are reallife processes that create and destroy pointlike charged particles, e.g. the annihilation of an electron and antielectron with the emission of two gamma rays. Physicists have, in fact, been struggling with infinities like this since about 1950, and the issue is far from resolved. Some theorists propose that apparently pointlike particles are actually not pointlike: close up, an electron might be like a little circular loop of string.

∫ *

13

The purpose of this problem is to find the force experienced by a straight, current-carrying wire running perpendicular to a uniform magnetic field. (a) Let A be the cross-sectional area of the wire, n the number of free charged particles per unit volume, q the charge per particle, and v the average velocity of the particles. Show that the current is I=Avnq. (b) Show that the magnetic force per unit length is AvnqB. (c) Combining these results, show that the force on the wire per unit length is equal to IB.

S

14

Suppose two long, parallel wires are carrying current I_{1} and I_{2}. The currents may be either in the same direction or in opposite directions. (a) Using the information from section 6.2, determine under what conditions the force is attractive, and under what conditions it is repulsive. Note that, because of the difficulties explored in problem 12 above, it's possible to get yourself tied up in knots if you use the energy approach of section 6.5. (b) Starting from the result of problem 13, calculate the force per unit length.

S

15

The figure shows cross-sectional views of two cubical capacitors, and a cross-sectional view of the same two capacitors put together so that their interiors coincide. A capacitor with the plates close together has a nearly uniform electric field between the plates, and almost zero field outside; these capacitors don't have their plates very close together compared to the dimensions of the plates, but for the purposes of this problem, assume that they still have approximately the kind of idealized field pattern shown in the figure. Each capacitor has an interior volume of 1.00 m^{3}, and is charged up to the point where its internal field is 1.00 V/m. (a) Calculate the energy stored in the electric field of each capacitor when they are separate. (b) Calculate the magnitude of the interior field when the two capacitors are put together in the manner shown. Ignore effects arising from the redistribution of each capacitor's charge under the influence of the other capacitor. (c) Calculate the energy of the put-together configuration. Does assembling them like this release energy, consume energy, or neither?

16

Section 6.2 states the following rule:

For a positively charged particle, the direction of the F vector is the one such that if you sight along it, the B vector is clockwise from the v vector.

Make a three-dimensional model of the three vectors using pencils or rolled-up pieces of paper to represent the vectors assembled with their tails together. Now write down every possible way in which the rule could be rewritten by scrambling up the three symbols F, B, and v. Referring to your model, which are correct and which are incorrect?

17

Prove that any two planar current loops with the same value of IA will experience the same torque in a magnetic field, regardless of their shapes. In other words, the dipole moment of a current loop can be defined as IA, regardless of whether its shape is a square.

*

18

A Helmholtz coil is defined as a pair of identical circular coils separated by a distance, h, equal to their radius, b. (Each coil may have more than one turn of wire.) Current circulates in the same direction in each coil, so the fields tend to reinforce each other in the interior region. This configuration has the advantage of being fairly open, so that other apparatus can be easily placed inside and subjected to the field while remaining visible from the outside. The choice of h=b results in the most uniform possible field near the center. (a) Find the percentage drop in the field at the center of one coil, compared to the full strength at the center of the whole apparatus. (b) What value of h (not equal to b) would make this percentage difference equal to zero?

19

(a) In the photo of the vacuum tube apparatus in section 6.2, infer the direction of the magnetic field from the motion of the electron beam.
(b) Based on your answer to a, find the direction of the currents in the coils. (c) What direction are the electrons in the coils going? (d) Are the currents in the coils repelling or attracting the currents consisting of the beam inside the tube? Compare with part a of problem 14.

20

In the photo of the vacuum tube apparatus in section 6.2, an approximately uniform magnetic field caused circular motion. Is there any other possibility besides a circle? What can happen in general?

*

21

In problem 1, you estimated the energy released in a bolt of lightning, based on the energy stored in the electric field immediately before the lightning occurs. The assumption was that the field would build up to a certain value, which is what is necessary to ionize air. However, real-life measurements always seemed to show electric fields strengths roughtly 10 times smaller than those required in that model. For a long time, it wasn't clear whether the field measurements were wrong, or the model was wrong. Research carried out in 2003 seems to show that the model was wrong. It is now believed that the final triggering of the bolt of lightning comes from cosmic rays that enter the atmosphere and ionize some of the air. If the field is 10 times smaller than the value assumed in problem 1, what effect does this have on the final result of problem 1?

22

In section 6.2 I gave an equation for the magnetic field in the interior of a solenoid, but that equation doesn't give the right answer near the mouths or on the outside. Although in general the computation of the field in these other regions is complicated, it is possible to find a precise, simple result for the field at the center of one of the mouths, using only symmetry and vector addition. What is it?