A spherical conductor of radius \(10~\text{cm}\) has a charge of \(3.2 \times 10^{-7}~\text{C}\) distributed uniformly. What is the magnitude of the electric field at a point \(15~\text{cm}\) from the center of the sphere? \(\frac{1}{4\pi \varepsilon _0} = 9\times 10^9~\text{N-m}^2/\text{C}^2\)
1. \(1.28\times 10^{5}~\text{N/C}\)
2. \(1.28\times 10^{6}~\text{N/C}\)
3. \(1.28\times 10^{7}~\text{N/C}\)
4. \(1.28\times 10^{4}~\text{N/C}\)

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

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

Four charges are arranged at the corners of a square ABCD as shown in the figure. The force on a positive charge kept at the center of the square is:

A point charge q is placed at the center of the open face of a hemispherical surface as shown in the figure. The flux linked with the surface is:

1.  zero

2.  $\frac{q}{2{\epsilon }_0}$

3.  $\frac{q}{{\epsilon }_0}$

4.  $q\pi r^2$

A charged particle q of mass m is released on the y-axis at y = a in an electric field $\overrightarrow{E}=-4y\hat{j}$. The speed of particle on reaching the origin will be:

1.  $\sqrt{\frac{2a}{mq}}$

2.  $\frac{a}{\sqrt{mq}}$

3.  $2a\sqrt{\frac{q}{m}}$

4.  $2\sqrt{\frac{a}{mq}}$

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

An electric dipole is placed at the centre of a sphere. Which of the following statements is correct?

An electric dipole is kept at the origin as shown in the diagram. The point A, B, C are on a circular arc with the centre of curvature at the origin. If the electric fields at A, B and C respectively are $\overrightarrow{E_1},$ $\overrightarrow{E_2},$ $\overrightarrow{E_3}$ , then which of the following is incorrect? \(\left ( d\gg l \right )\)

                
1.  $\overrightarrow{E_1}$ $=$ $-\overrightarrow{E_3}$
2.  $\overrightarrow{E_1}$ $=$ $-2\overrightarrow{E_2}$
3.  $\overrightarrow{E_1}$ $=$ $\overrightarrow{E_3}$
4.  $\overrightarrow{E_3}$ $=$ $-\overrightarrow{2E_2}$