Electrostatics

Within a spherical charge distribution of charge density ρ(r), N equipotential surfaces of potential $V_0$, $V_0+\Delta V$, $V_0+2\Delta V$, .......... $V_0+N\Delta V$ (ΔV > 0), are drawn and have increasing radii $r_0$, $r_1$, $r_2$, .......... $r_N$, respectively. If the difference in the radii of the surfaces is constant for all values of $V_0$ and ΔV then :

Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed '$v$' in a uniform magnetic field B going into the plane of the paper (See figure). If charge densities $\sigma_1$ and $\sigma_2$ are induced on the left and right surfaces, respectively, of the sheet then (ignore fringe effects) :

An infinitely long uniform line charge distribution of charge per unit length $\lambda$ lies parallel to the $y$-axis in the $y$-$z$ plane at $z=\dfrac{\sqrt3}{2}a$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $ABCD$ lying in the $x$-$y$ plane with its centre at the origin is $\dfrac{\lambda L}{n\varepsilon_0}$ ($\varepsilon_0$ = permittivity of free space), then the value of $n$ is

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\lambda$ are kept parallel to each other. In their resulting electric field, point charges $q$ and $-q$ are kept in equilibrium between them. The point charges are confined to move in the $x$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is(are)

Consider a uniform spherical charge distribution of radius $R_1$ centred at the origin O. In this distribution, a spherical cavity of radius $R_2$, centred at $P$ with distance $\mathrm{OP}=a=R_1-R_2$ (see figure) is made. If the electric field inside the cavity at position $\vec r$ is $\vec E(\vec r)$, then the correct statement(s) is(are)

A long cylindrical shell carries positive surface charge σ in the upper half and negative surface charge −σ in the lower half. The electric field lines around the cylinder will look like figure given in :

(figures are schematic and not drawn to scale)

A uniformly charged solid sphere of radius R has potential $V_0$ (measured with respect to ∞) on its surface. For this sphere the equipotential surfaces with potentials $\dfrac{3V_0}2$, $\dfrac{5V_0}4$, $\dfrac{3V_0}4$ and $\dfrac{V_0}4$ have radius $R_1$, $R_2$, $R_3$ and $R_4$ respectively. Then

A thin disc of radius b = 2a has a concentric hole of radius 'a' in it (see figure). It carries uniform surface charge 'σ' on it. If the electric field on its axis at height 'h' (h<<a) from its centre is given as 'Ch' then value of 'C' is :

Shown in the figure are two point charges +Q and −Q inside the cavity of a spherical shell. The charges are kept near the surface of the cavity on opposite sides of the centre of the shell. If $\sigma_1$ is the surface charge on the inner surface and $Q_1$ net charge on it and $\sigma_2$ the surface charge on the outer surface and $Q_2$ net charge on it then :

A wire, of length L(=20 cm), is bent into a semi-circular arc. If the two equal halves, of the arc, were each to be uniformly charged with charges ±Q, [|Q| = $10^3\epsilon_0$ Coulomb where $\epsilon_0$ is the permittivity (in SI units) of free space] the net electric field at the centre O of the semi-circular arc would be :

An electric field $\vec E=\left(25\hat i+30\hat j\right)$ NC$^{-1}$ exists in a region of space. If the potential at the origin is taken to be zero then the potential at $x=2$ m, $y=2$ m is :

Let $E_1(r)$, $E_2(r)$ and $E_3(r)$ be the respective electric fields at a distance $r$ from a point charge $Q$, an infinitely long wire with constant linear charge density $\lambda$, and an infinite plane with uniform surface charge density $\sigma$. If $E_1(r_0)=E_2(r_0)=E_3(r_0)$ at a given distance $r_0$, then

Charges $Q$, $2Q$ and $4Q$ are uniformly distributed in three dielectric solid spheres 1, 2 and 3 of radii $R/2$, $R$ and $2R$, respectively, as shown in figure. If magnitudes of the electric fields at point $P$ at a distance $R$ from the center of spheres 1, 2 and 3 are $E_1$, $E_2$ and $E_3$ respectively, then

Four charges $Q_1$, $Q_2$, $Q_3$ and $Q_4$ of same magnitude are fixed along the $x$ axis at $x=-2a$, $-a$, $+a$ and $+2a$, respectively. A positive charge $q$ is placed on the positive $y$ axis at a distance $b>0$. Four options of the signs of these charges are given in List I. The direction of the forces on the charge $q$ is given in List II. Match List I with List II and select the correct answer using the code given below the lists.

Assume that an electric field $\vec E=30x^2\,\hat i$ exists in space. Then the potential difference $V_A-V_O$, where $V_O$ is the potential at the origin and $V_A$ the potential at $x=2$ m is :

The magnitude of the average electric field normally present in the atmosphere just above the surface of the Earth is about 150 N/C, directed inward towards the center of the Earth. This gives the total net surface charge carried by the Earth to be :

[Given $\epsilon_o=8.85\times10^{-12}$ C$^2$/N-m$^2$, $R_E=6.37\times10^6$ m]

A cone of base radius R and height $h$ is located in a uniform electric field $\vec E$ parallel to its base. The electric flux entering the cone is :

A spherically symmetric charge distribution is characterised by a charge density having the following variation :
$$\rho(r)=\rho_o\left(1-\frac rR\right)\ \text{for } r<R,\qquad\rho(r)=0\ \text{for } r\ge R$$
Where r is the distance from the centre of the charge distribution and $\rho_o$ is a constant. The electric field at an internal point (r < R) is :

The electric field in a region of space is given by, $\vec E=E_o\hat i+2E_o\hat j$ where $E_0=100$ N/C. The flux of this field through a circular surface of radius 0.02 m parallel to the Y-Z plane is nearly :

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 $\dfrac{\rho_1}{\rho_2}$ can be