Two point dipoles of dipole moment ${\overrightarrow{\mathrm{p}}}_1$ and ${\overrightarrow{\mathrm{p}}}_2$ are at a distance x from each other and ${\overrightarrow{\mathrm{p}}}_1\left|\right|{\overrightarrow{\mathrm{p}}}_2$. The force between the dipole is:

1. $\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{4p_1{p}_2}{x^4}$

2. $\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{3p_1{p}_2}{x^3}$

3. $\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{6p_1{p}_2}{x^4}$

4. $\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{8p_1{p}_2}{x^4}$

If there were only one type of charge in the universe, then,

Refer to the arrangement of charges in the figure and a Gaussian surface of radius R with Q at the centre. Then:

Choose the correct statement(s): 
1.  a and d
2.  a and c
3.  b and d
4.  c and d

The electric field at a point on the equatorial plane at a distance \(r\) from the centre of a dipole having dipole moment $\stackrel{}{}$\(\overrightarrow{P}\) is given by:
(\(r\gg\) separation of two charges forming the dipole, \(\epsilon_{0} =\) permittivity of free space) 
1. \(\overrightarrow{E}=\frac{\overrightarrow{P}}{4\pi \epsilon _{0}r^{3}}\) 

2. \(\overrightarrow{E}=\frac{2\overrightarrow{P}}{\pi \epsilon _{0}r^{3}}\)

3. \(\overrightarrow{E}=-\frac{\overrightarrow{P}}{4\pi \epsilon _{0}r^{2}}\)

4. \(\overrightarrow{E}=-\frac{\overrightarrow{P}}{4\pi \epsilon _{0}r^{3}}\)

The acceleration of an electron due to the mutual attraction between the electron and a proton when they are \(1.6~\mathring{A}\) apart is:
( \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9~ \text{Nm}^2 \text{C}^{-2}\) )

The figure shows electric field lines in which an electric dipole p is placed as shown. Which of the following statements is correct?
                      

A hollow metal sphere of radius \(R\) is uniformly charged. The electric field due to the sphere at a distance \(r\) from the centre:

In a certain region of space, the electric field is along the z-direction throughout. The magnitude of the electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 10⁵ NC^-1 per meter. What is the torque experienced by a system having a total dipole moment equal to ${10}^{-7}$ $\mathrm{C-}m$ in the negative z-direction? 

A polythene piece rubbed with wool is found to have a negative charge of $3\times {10}^{-7} C$. Transfer of mass from wool to polythene is:
1. 0.7 × 10 ^{- 18}   kg
2. 1.7 × 10 ^{- 17}   kg
3. 0.7 × 10 ^{- 17}   kg
4. 1.7 × 10 ^{- 18}   kg

A charge \(q\) is placed in a uniform electric field \(E.\) If it is released, then the kinetic energy of the charge after travelling distance \(y\) will be:
1. \(qEy\)
2. \(2qEy\)
3. $\frac{qEy}{2}$
4. $\sqrt{qEy}$