A cylindrical solid rod of length $L$ and radius $a$ is connected across a source of emf $V$ and negligible internal resistance as shown in the figure. The resistivity of the rod at point $P$ at a distance $x$ from the left end is given by $\rho =bx$ (where $b$ is a positive constant). The electric field at point $P$ is $\left(\frac{nV}{L^2}\right)x$. The value of $n$ is

There are six parallel plates of equal area A are arranged as shown in figure $(\mathrm{d}<<\mathrm{A}).$ The equivalent capacitance between points $2$ and $5,$ is $\alpha \frac{{ϵ}_0 A}{ d}.$ Then find the value of $\alpha$

In the arrangement shown in figure, find the potential difference $V_B-V_A$ (in $V$ ). (Take $V_0=55 \mathrm{V}$)

A coil of self inductance $L=1 \mathrm{H}$ & resistance $r=4 \mathrm{\Omega }$ is connected in series with a variable resistance $R$ and an a.c. source with $\omega =3 \mathrm{rad} {\mathrm{s}}^{-1}$ as shown in the figure. What should be the value of $R$ (in $\mathrm{\Omega }$) so that the rate of heat loss through $R$ is maximum?