Three identical positive point charges, as shown are placed at the vertices of an isosceles right-angled triangle. Which of the numbered vectors coincides in direction with the electric field at the mid-point \(M\) of the hypotenuse?
                 
1. \(1\)
2. \(2\)
3. \(3\)
4. \(4\)

The figures below show regular hexagons, with charges at the vertices. In which of the following cases the electric field at the centre is not zero?

(1) 1

(2) 2

(3) 3

(4) 4

An electron enters an electric field with its velocity in the direction of the electric lines of force. Then: 

Two small spherical balls each carrying a charge Q = 10 μC (10 micro-coulomb) are suspended by two insulating threads of equal lengths 1m each, from a point fixed in the ceiling. It is found that in equilibrium threads are separated by an angle 60° between them, as shown in the figure. What is the tension in the threads (Given: $\frac{1}{\left(4\pi {\epsilon }_0\right)}=9\times {10}^{9}Nm/C^2$) 

(1) 18 N

(2) 1.8 N

(3) 0.18 N

(4) None of the above

An electron having charge \(e\) and mass \(m\) is moving in a uniform electric field \(E.\) Its acceleration will be:
1. \(\dfrac{e^2}{m}\)
2. \(\dfrac{E^2e}{m}\)
3. \(\dfrac{eE}{m}\)
4. \(\dfrac{mE}{e}\)

If an insulated non-conducting sphere of radius R has charge density ρ. The electric field at a distance r from the centre of sphere (r < R) will be 

(1) $\frac{\rho R}{3{\epsilon }_0}$

(2) $\frac{\rho r}{{\epsilon }_0}$

(3) $\frac{\rho r}{3{\epsilon }_0}$

(4) $\frac{3\rho R}{{\epsilon }_0}$ 

Infinite charges of magnitude q each are lying at x =1, 2, 4, 8... meter on X-axis. The value of the intensity of the electric field at point x = 0 due to these charges will be 

(1) 12 × 10⁹q N/C

(2) Zero

(3) 6 × 10⁹q N/C

(4) 4 × 10⁹q N/C

A pendulum bob of mass $30.7\times {10}^{-6}kg$ and carrying a charge $2\times {10}^{-8}C$ is at rest in a horizontal uniform electric field of 20000 V/m. The tension in the thread of the pendulum is $(g=9.8m/s^2)$

(1) $3\times {10}^{-4}N$

(2) $4\times {10}^{-4}N$

(3) $5\times {10}^{-4}N$

(4) $6\times {10}^{-4}N$ 

A charged ball \(B\) hangs from a silk thread \(S,\) which makes an angle \(\theta\) with a large charged conducting sheet \(P,\) as shown in the figure. The surface charge density \(\sigma\) of the sheet is proportional to: 

1. \(\sin\theta\)
 2. \(\tan\theta\)
 3. \(\cos\theta\)
 4. \(\cot\theta\)

Two-point charges +8q and –2q are located at x = 0 and x = L respectively. The location of a point on the x-axis at which the net electric field due to these two point charges is zero is 

1. 8 L

2. 4 L

3. 2 L

4. $\frac{L}{4}$