How does the relation $\Delta V=i R$ apply to resistors that do not obey Ohm's law?

The copper windings of a motor have a resistance of $50 \Omega$ at $20^{\circ} \mathrm{C}$ when the motor is idle. After running for several hours the resistance rises to $58 \Omega$. What is the temperature of the windings? Ignore changes in the dimensions of the windings.

Explain why the momentum that conduction electrons transfer to the ions in a metal conductor does not give rise to a resultant force on the conductor.

Show that, according to the free-electron model of electrical conduction in metals and classical physics, the resistivity of metals should be proportional to $\sqrt{T}$, where $T$ is absolute temperature. (Hint: Treat the electrons as an ideal gas.)

Will the drift velocity of electrons in a current-carrying metal conductor change when the temperature of the conductor is increased? Explain.

A resistor is in the shape of a spherical shell, with an inside surface of radius of $a$ covered with a conducting material and an outside surface of radius $b$ covered with a conducting material. Assuming a uniform resistivity $\rho$, calculate the resistance between the conducting surfaces.

A resistor is in the shape of a truncated right circular cone. The end radii are $a$ and $b$, and the length is $L$. If the taper is small, we may assume that the current density is uniform across any cross section. (a) Calculate the resistance of this object. (b) Show that your answer reduces to $\rho L / A$ for the special case of zero taper $(a=b)$.

Discuss the difficulties of testing whether the filament of a lightbulb obeys Ohm's law.

Sliding across the seat of an automobile can generate potentials of several thousand volts. Why is the slider not electrocuted?

It is desired to make a long cylindrical conductor whose temperature coefficient of resistivity at $20^{\circ} \mathrm{C}$ will be close to zero. If such a conductor is made by assembling alternate disks of iron and carbon, find the ratio of the thickness of a carbon disk to that of an iron disk. (For carbon, $\rho=3500 \mathrm{x}$ $10^{-8} \Omega \cdot \mathrm{m}$ and $\alpha=-0.50 \times 10^{-3} / \mathrm{C}^0$.)

A steel trolley-car rail has a cross-sectional area of $56 \mathrm{~cm}^2$. What is the resistance of $11 \mathrm{~km}$ of rail? The resistivity of the steel is $3.0 \times 10^{-7} \mathrm{\Omega} \cdot \mathrm{m}$

A human being can be electrocuted if a current as small as $50 \mathrm{~mA}$ passes near the heart. An electrician working with sweaty hands makes good contact with two conductor buing laid one in each hand. If the electrician's resistance is $1800 \Omega$, what might be the fatal voltage difference? (Electricians often work with "live" wires.)

When a metal rod is heated. not only its resistance but also its length and its cross-sectional area change. The relation $R=$ $\rho L / A$ suggests that all three factors should be taken into $3 c$ count in measuring $\rho$ at various temperatures. (a) If the temperature changes by $1.0 \mathrm{C}^{\circ}$, what fractional changes in $R, L$, and $A$ occur for a copper conductot? (b) What conclusion do you draw? The coefficient of linear expansion is $1.7 \times$ $10^{-5} / \mathrm{C}^{\circ}$.

How would you measure the resistance of a pretzel-shape block of metal? Give specific details to clarify the concept.