The resistivity of a cylindrical conductor carrying steady current along its length varies linearly with the distance from the current carrying end as given by $\rho ={\rho }_0\left(1+\frac{x}{l}\right)$ where $l$ is the length of the conductor and $x$ is the distance from the current entry end. ${\rho }_o$ is a positive constant.

Point charges $+q$ and $-q$ are located at the vertices of a square with diagonals $2l$ as shown in the figure. The magnitude of the electric field strength at a point, located symmetrically with respect to the vertices of the square at a distance $x$ from its center is

A thin disc of radius $b=2a$ has a concentric hole of radius $a$ in it (see figure). It carries uniform surface charge $\sigma$  on it. If the electric field on its axis at a height $\text{h}\left(\text{h}<<\text{a}\right)$ from its centre is given as $\text{C}\text{h}$ then the value of $C$ is

Two mutually perpendicular infinitely long straight conductors carrying uniformly distributed charges of linear densities ${\lambda }_1$ and ${\lambda }_2$ are positioned at a distance $r$ from each other.

Force between the conductors depends on $r$ as

The bob of a pendulum of mass $m$ suspended by an inextensible string of length $L$ as shown in the figure carries a small charge $q$. An infinite horizontal plane conductor with uniform surface charge density $\sigma$ is placed below it. What will be the time period of the pendulum for small amplitude oscillations?

A current of $10 \mathrm{A}$ is passing through a long wire which has a semicircular loop of the radius $20 \mathrm{cm}$ as shown in the figure. Magnetic field produced at the centre of the loop is:

Find the electric field at point $P$ (as shown in figure) on the perpendicular bisector of a uniformly charged thin wire of length $L$ carrying a charge $Q$. The distance of the point $P$ from the centre of the rod is $a=\frac{\sqrt{3}}{2}L$

A cube of side $a$ has point charges $+\mathrm{Q}$ located at each of its vertices except at the origin where the charge is $-\mathrm{Q}.$ The electric field at the centre of cube is:

Two non-conducting solid spheres of radii $R$ and $2R$, having uniform volume charge densities ${\rho }_1$ and ${\rho }_2$, respectively, touch each other. The net electric field at a distance $2R$ from the centre of the smaller sphere, along the line joining the centres of the spheres, is zero. The ratio $\frac{{\rho }_1}{{\rho }_2}$ can be