An electric dipole of moment p is placed in an electric field of intensity E. The dipole acquires a position such that the axis of the dipole makes an angle $\theta$ with the direction of the field. Assuming that the potential energy of the dipole to be zero when $\theta =90°$, the torque and the potential energy of the dipole will respectively be

1. $pE \sin \theta ,-pE \cos \theta$

2. $pE \sin \theta ,-2pE \cos \theta$

3. $pE \sin \theta ,2pE \cos \theta$

4. $pE \cos \theta ,-pE \sin \theta$

What is the flux through a cube of side 'a' if a point charge q is at one of its corners?

1. $\frac{2q}{{\epsilon }_0}$                                         

2. $\frac{q}{8{\epsilon }_0}$

3. $\frac{q}{{\epsilon }_0}$                                         

4. $\frac{q}{2{\epsilon }_0}6a^2$

A charge Q is enclosed by a Gaussian spherical surface of radius R. If the radius is doubled, then the outward electric flux will

1. be reduced to half                                 

2. remain the same

3. be doubled

4. increase four times

Two positive ions, each carrying a charge q, are separated by a distance d. If F is the force of repulsion between the ions, the number of electrons missing from each ion will be (e being the charge on an electron)

1. $\frac{4{\mathrm{\pi \epsilon }}_{\circ }{\mathrm{Fd}}^2 }{e^2}$                  2. $\sqrt{\frac{4{\mathrm{\pi \epsilon }}_{\circ }{\mathrm{Fe}}^2}{d^2}}$

3. $\sqrt{\frac{4{\mathrm{\pi \epsilon }}_{\circ }{\mathrm{Fd}}^2}{e^2}}$               4. $\frac{4{\mathrm{\pi \epsilon }}_{\circ }{\mathrm{Fd}}^2}{q^2}$

A surface of side L metre in the plane of the paper is placed in a uniform electric field E(volt/m) acting along the same plane at an angle $\theta$ with the horizontal side of the square as shown in figure. The electric flux linked to the surface in unit of V-m, is

(1) EL² 

(2) EL²cos$\theta$

(3) EL²sin$\theta$

(4) 0

The electric field at a distance $\frac{3\mathrm{R}}{2}$ from the centre of a charged conducting spherical shell of radius R is E. The electric field at a distance $\frac{\mathrm{R}}{2}$ from the centre of the sphere is 

1. $\mathrm{zero}$                                     2. $E$

3. $\frac{E}{2}$                                       4. $\frac{E}{3}$

A thin conducting ring of radius R is given a charge +Q. The electric field at the centre O of the ring due to the charge on the part AKB of the ring is E. The electric field at the centre due to the charge on the part ACDB of the ring is 

(1) 3E along KO 

(2) E along OK 

(3) E along KO

(4) 3E along OK  

Two pith balls carrying equal charges are suspended from a common point by strings of equal length, the equilibrium separation between them is r. Now the strings are rigidly clamped at half the height. The equilibrium separation between the balls now become:

1. (1/√2)²

2. $(r/\sqrt[3]{2})$

3. (2r/√3)

4. (2r/3)

The electric field in a certain region is acting radially outward and is given by E=Ar. A charge contained in a sphere of radius 'a' centered at the origin of the field will be given by

1. $4\pi {\in }_{o}A{a}^2$

2. $\pi {\in }_{o}A{a}^2$

3. $4\pi {\in }_{o}A{a}^3$

4. ${\in }_{o}A{a}^2$

Two identical charged spheres suspended from a common point by two massless strings of lengths l are initially at a distance d(d < < l) apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity v. Then, v varies as a function of the distance x between the sphere, as:

(a) \(v \propto x\)

(b) \(v \propto x^{\frac{-1}{2}}\)

(c) \(v \propto x^{-1}\)

(d) \(v \propto x^{\frac{1}{2}}\)${\frac{\mathrm{}}{}}^{}$