In the given figure, a metal rod is forced to move with constant velocity $\overrightarrow{v}$ along two parallel metal rails, connected with a strip of metal at one end. A magnetic field of magnitude, $B=0.125 \mathrm{T}$ points out of the page. $\left(\mathrm{a}\right)$ If the rails are separated by, $L=25.0 \mathrm{cm}$ and the speed of the rod is $38.0 \mathrm{cm} {\mathrm{s}}^{-1},$ what emf is generated? $\left(b\right)$ If the rod has a resistance of $18.0 \mathrm{\Omega }$ and the rails and connector have negligible resistance, what is the current in the rod? $\left(\mathrm{c}\right)$ At what rate is energy being transferred to thermal energy? $\left(d\right)$ What is the magnitude of the leftward force that causes the rod to move?

The conducting rod has length $L$ and is being pulled along horizontal, frictionless conducting rails at a constant velocity $\overrightarrow{v}$. The rails are connected at one end with a metal strip. A uniform magnetic field $\overrightarrow{B}\text{,}$ directed out of the page, fills the region in which the rod moves. Assume that $L=10 \mathrm{cm}\text{,} v=5.0 \mathrm{m} {\mathrm{s}}^{-1}$ and $B=2.4 \mathrm{T}$. What are the (a) magnitude and (b) direction (up or down the page) of the emf induced in the rod? What are the (c) size and (d) direction of the current in the conducting loop? Assume that the resistance of the rod is $0.40 \mathrm{\Omega }$ and that the resistance of the rails and metal strip is negligibly small. (e) At what rate is thermal energy being generated in the rod? (f) What external force on the rod is needed to maintain $\overrightarrow{v}\text{?}$ (g) At what rate does this force do work on the rod? (h) What is the magnitude of the magnetic force on the rod? 

A coil $C$ of $N$ turns is placed around a long solenoid $S$ of radius $R$ and $n$ turns per unit length, as shown in the given figure. $\left(a\right)$ Show that the mutual inductance for the coil-solenoid combination is given by, $M={\mu }_0\pi R^{2}nN$. $\left(\mathrm{b}\right)$ Explain why $M$ does not depend on the shape, size or possible lack of close packing of the coil.  

If $50.0 \mathrm{cm}$ of copper wire (diameter$=3.00 \mathrm{mm}$) is formed into a circular loop and placed perpendicular to a uniform magnetic field that is increasing at the constant rate of $30.0 \mathrm{mT} {\mathrm{s}}^{-1}$, at what rate is thermal energy generated in the loop? (Resistivity of copper wire is $1.69\times {10}^{-8} \mathrm{\Omega } \mathrm{m}$)

Figure shows, a copper strip of width $W=20.0 \mathrm{cm}$ that has been bent to form a shape that consists of a tube of radius $R=1.8 \mathrm{cm}$ plus two parallel flat extensions. Current $i=35 \mathrm{mA}$ is distributed uniformly across the width so that the tube is effectively a one-turn solenoid. Assume that the magnetic field outside the tube is negligible and the field inside the tube is uniform. What are $\left(\mathrm{a}\right)$ the magnetic field magnitude inside the tube and $\left(\mathrm{b}\right)$ the inductance of the tube (excluding the flat extensions)? 

Figure shows two parallel loops of wire having a common axis. The smaller loop (radius $r$) is above the larger loop (radius $R$) by a distance $x\gg R.$ Consequently, the magnetic field due to the counter clockwise current $i$ in the larger loop is nearly uniform throughout the smaller loop. Suppose that $x$ is increasing at the constant rate $\frac{\mathrm{d}x}{\mathrm{d}t}=v$. (a) Find an expression for the magnetic flux through the area of the smaller loop as a function of $x$. In the smaller loop, find (b) an expression for the induced emf and (c) the direction of the induced current. 

Suppose the emf of the battery, in the circuit shown in figure, varies with time $t$ so that the current is given by $i\left(t\right)=3.0+5.0t$, where $i$ is in $\mathrm{A}$ and $t$ is in $\mathrm{s}$. Take $R=8.0 \mathrm{\Omega }$ and $L=6.0 \mathrm{H},$ and find an expression for the battery emf as a function of $t$.