Figure 6.88 shows a long, rectangular, conducting loop of width $I$, mass $m$ and resistance $R$ placed partly in a perpendicular magnetic field $B$. With what velocity should it be pushed downwards so that it may continue to fall without any acceleration?

 

Figure 6.89 shows a wire of mass m and length $l$ which can slide on a pair of parallel, smooth, horizontal rails placed in a vertical magnetic field $B$. The rails are connected by a capacitor of capacitance $C$. The electric resistance of the rails and the wire is zero. If a constant force F acts on the wire as shown in the figure, find the acceleration of the wire.

A rectangular wire frame, as shown in Fig. 6.90(a) is placed in uniform magnetic field directed upward and normal to the plane of the paper. The part $AB$ is connected to a spring. The spring is stretched and released when the wire $AB$ has come to the position $A\text{'} B\text{'} (t=0)$. Explain qualitatively how induced emf in the coil would vary with time. Neglect damping of oscillations of the spring.

Figure 6.91 shows two long coaxial solenoids, each of length '$l$'. The outer solenoid has an area of cross-section $A_1$ and number of turns/length $n_1$. The corresponding values for the inner solenoid are $A_2$ and $n_2$. Write the expression for self inductance $L_1$, $L_2$ of the two coils and their mutual inductance $M$. Hence show that $M<\sqrt{L_1{L}_2}$.

The magnetic flux passing perpendicular to the plane of a coil and directed into the paper is given by $\varphi =5t^2+6t+2$, where $\varphi$ is in milliwebers and $t$ in seconds.

What is the direction of current through the resistor $R$?

A square metal wire loop of side $10 \mathrm{cm}$ and resistance $1 \mathrm{ohm}$ is moved with a constant velocity $v_0$ in a uniform magnetic field of induction $B=2 \mathrm{Wb} {\mathrm{m}}^{-2}$ as shown in Fig. 6.94. The magnetic lines are perpendicular to the plane of the loop (directed into the paper). The loop is connected to a network of resistors each of value $3 \mathrm{\Omega }$. The resistances of the loop wires $OS$ and $PQ$ are negligible. What should be the speed of the loop so as to have a steady current of $1 \mathrm{mA}$ in the loop? Give the direction of the current in the loop?

Refer to Fig. 6.95. The arm $PQ$ of the rectangular conductor is moved from $x=0$ to the right side. The uniform magnetic field is perpendicular to the plane and extends from $x=0$ to $x=b$ and is zero for $x>b$. Only the arm $PQ$ possesses substantial resistance $r$. Consider the situation when the arm $PQ$ is pulled outwards from $x=0$ to $x=2 \mathrm{b}$ and is then moved back to $x=0$ with constant speed $v$. Obtain expressions for the flux, the induced emf, the force necessary to pull the arm and the power dissipated as Joule heat. Sketch the variation of these quantities with time.