Electrostatics

Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,

Consider a thin spherical shell of radius $R$ with its centre at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field $|\vec E(r)|$ and the electric potential $V(r)$ with the distance $r$ from the centre, is best represented by which graph?

Two large vertical and parallel metal plates having a separation of 1 cm are connected to a DC voltage source of potential difference $X$. A proton is released at rest midway between the two plates. It is found to move at 45° to the vertical JUST after release. Then $X$ is nearly

A cubical region of side $a$ has its centre at the origin. It encloses three fixed point charges, $-q$ at $(0,-a/4,0)$, $+3q$ at $(0,0,0)$ and $-q$ at $(0,+a/4,0)$. Choose the correct option(s).

An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho$. It has a spherical cavity of radius $R/2$ with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point $P$, which is at a distance $2R$ from the axis of the cylinder, is given by the expression $\dfrac{23\rho R}{16k\varepsilon_0}$. The value of $k$ is

A proton is fired from very far away towards a nucleus with charge $Q=120e$, where $e$ is the electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm) of the proton at its start is: (take the proton mass, $m_p=(5/3)\times10^{-27}$ kg; $h/e=4.2\times10^{-15}$ J.s/C; $\dfrac1{4\pi\varepsilon_0}=9\times10^9$ m/F; 1 fm $=10^{-15}$ m)

Six point charges are kept at the vertices of a regular hexagon of side $L$ and centre $O$, as shown in the figure. Given that $K=\dfrac1{4\pi\varepsilon_0}\dfrac q{L^2}$, which of the following statement(s) is(are) correct?

Consider an electric field $\vec E=E_0\hat x$, where $E_0$ is a constant. The flux through the shaded area (as shown in the figure) due to this field is

A spherical metal shell A of radius $R_A$ and a solid metal sphere B of radius $R_B$ ($<R_A$) are kept far apart and each is given charge '$+Q$'. Now they are connected by a thin metal wire. Then

Four point charges, each of $+q$, are rigidly fixed at the four corners of a square planar soap film of side '$a$'. The surface tension of the soap film is $\gamma$. The system of charges and planar film are in equilibrium, and $a=k\left[\dfrac{q^2}\gamma\right]^{1/N}$, where '$k$' is a constant. Then $N$ is

A wooden block performs SHM on a frictionless surface with frequency, $\nu_0$. The block carries a charge $+Q$ on its surface. If now a uniform electric field $\vec E$ is switched-on as shown, then the SHM of the block will be

Which of the following statement(s) is/are correct?

A silver sphere of radius 1 cm and work function 4.7 eV is suspended from an insulating thread in free-space. It is under continuous illumination of 200 nm wavelength light. As photoelectrons are emitted, the sphere gets charged and acquires a potential. The maximum number of photoelectrons emitted from the sphere is $A\times10^{Z}$ (where $1<A<10$). The value of '$Z$' is

A few electric field lines for a system of two charges $Q_1$ and $Q_2$ fixed at two different points on the x-axis are shown in the figure. These lines suggest that

A tiny spherical oil drop carrying a net charge $q$ is balanced in still air with a vertical uniform electric field of strength $\dfrac{81\pi}{7}\times10^5$ V m⁻¹. When the field is switched off, the drop is observed to fall with terminal velocity $2\times10^{-3}$ m s⁻¹. Given $g=9.8$ m s⁻², viscosity of the air $=1.8\times10^{-5}$ N s m⁻² and the density of oil $=900$ kg m⁻³, the magnitude of $q$ is

A uniformly charged thin spherical shell of radius $R$ carries uniform surface charge density of $\sigma$ per unit area. It is made of two hemispherical shells, held together by pressing them with force $F$ (see figure). $F$ is proportional to

Three concentric metallic spherical shells of radii $R$, $2R$, $3R$, are given charges $Q_1$, $Q_2$, $Q_3$, respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, $Q_1:Q_2:Q_3$, is

A disk of radius $a/4$ having a uniformly distributed charge $6C$ is placed in the $x$-$y$ plane with its centre at $(-a/2,0,0)$. A rod of length $a$ carrying a uniformly distributed charge $8C$ is placed on the $x$-axis from $x=a/4$ to $x=5a/4$. Two point charges $-7C$ and $3C$ are placed at $(a/4,-a/4,0)$ and $(-3a/4,3a/4,0)$, respectively. Consider a cubical surface formed by six surfaces $x=\pm a/2$, $y=\pm a/2$, $z=\pm a/2$. The electric flux through this cubical surface is

Assume that two deuteron nuclei in the core of fusion reactor at temperature $T$ are moving towards each other, each with kinetic energy $1.5\,kT$, when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature $T$ required for them to reach a separation of $4\times10^{-15}$ m is in the range

Six point charges, each of the same magnitude $q$, are arranged in different manners as shown in Column II. In each case, a point M and a line PQ passing through M are shown. Let $E$ be the electric field and $V$ be the electric potential at M (potential at infinity is zero) due to the given charge distribution when it is at rest. Now, the whole system is set into rotation with a constant angular velocity about the line PQ. Let $B$ be the magnetic field at M and $\mu$ be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current.