S S Krotov Solutions for Chapter: Electricity and Magnetism, Exercise 1: Exercise

A long cylindrical coil of inductance $L_1$ is wound on a bobbin of diameter $D_1$. The magnetic induction in the coil connected to a current source is $B_1$. After rewinding the coil on a bobbin of diameter $D_2$, its inductance becomes $L_2$.

Determine the magnetic induction $B_2$ of the field in the new coil connected to the same current source, assuming that the length of the wire is much larger than that of the coil.

Two long cylindrical coils with uniform windings of the same length and nearly the same radius have inductances $L_1$ and $L_2$. The coils are coaxially inserted into each other and connected to a current source as shown in. The directions of the current in the circuit and in the turns are shown by arrows.

Determine the inductance $L$ of such a composite coil.

A winch is driven by an electric motor with a separate excitation and fed from a battery of emf $\epsilon =300 \mathrm{V}$. The rope and the hook of the winch rise at a velocity $v_1=4{\mathrm{ms}}^{-1}$ without a load and at a velocity $v_2=1{\mathrm{ms}}^{-1}$ with a load of mass $m=10\mathrm{kg}$.

Determine the velocity $v^{\text{'}}$ of the load and its mass $m^{\text{'}}$ for which the winch has the maximum power, neglecting the mass of the rope and the hook.

A perfect diode is connected to an a.c. circuit.

Determine the limits within which the voltage between the anode and the cathode varies.

A capacitor of unknown capacitance, a coil of inductance $L$, and a resistor of resistance $R$ are connected to a source of a.c. voltage $\epsilon ={\epsilon }_0\cos \omega t$. The current in the circuit is $I=\left(\frac{E_0}{R}\right)\cos \omega t$ .

Determine the amplitude $U_0$ of the voltage across the capacitor plates.

Under the action of a constant voltage $U$, a capacitor of capacitance $C={10}^{-11}\mathrm{F}$

included in the circuit shown in Fig. is charged to $q_1={10}^{-9}C$. The inductance of the coill is $L={10}^{-5}\mathrm{H}$, and the resistance of the resistor is $R=100\mathrm{\Omega }$.

Determine the amplitude $q_0$ of steadystate oscillations of the charge on the capacitor at resonance if the amplitude of the externàl sinusoidal voltage is $U_n=U$.

A bank of two series-connected capacitors of capacitance $C$ each is charged to a voltage $U$ and is connected to a coil of inductance $L$ so that an oscillatory circuit is formed at the initial moment. After a time $\tau$, a breakdown occurs in one of the capacitors, and the resistance between its plates becomes zero.

Determine the amplitude $q_0$ of charge oscillations on the undamaged capacitor.

How can the damage due to overheating the coil of a superconducting solenoid be avoided?