A long cylindrical shell is uniformly charged such that the upper half of its curved surface carries a positive surface charge density \(\sigma,\) while the lower half carries an equal negative surface charge density \(-\sigma.\) Which of the following schematic diagrams (not drawn to scale) most accurately represents the resulting pattern of electric field lines around the cylinder?

1.
2.
3.
4.

The region between two concentric spheres of radii '\(a\)' and '\(b\)' , respectively (see figure), has volume charge density \(\rho=\frac{A}{r}\), where \(A\) is a constant and \(r\) is the distance from the centre. At the centre of the spheres is a point charge \(Q\). The value of \(A\) such that the electric field in the region between the spheres will be constant, is:
             
1. \( \frac{Q}{2 \pi a^2} \)
2. \(\frac{Q}{2 \pi\left(b^2-a^2\right)} \)
3. \(\frac{2 Q}{\pi\left(a^2-b^2\right)} \)
4. \(\frac{2 Q}{\pi a^2}\)

An electric dipole with dipole moment \(\vec p,\) inclined at an angle \(\theta\) to the \(x\text-\)axis, experiences a torque \(\vec{\tau_1}= \tau \hat k\) in the field \(\vec E_1= E \hat i,\) and a torque \(\vec{\tau_2}=-\vec{\tau_1}\)​ in the field \(\vec E_2 = \sqrt{3}E \hat j.\) The angle \(\theta\) is:

The bob of a simple pendulum has mass \(2~\text{g}\) and a charge of \(5.0~\mu \text{C}\). It is at rest in a uniform horizontal electric field of intensity \(2000~\text{V/m}\). At equilibrium, the angle that the pendulum makes with the vertical is: (take \(g = 10~\text{m/s}^2\)${\mathrm{}}^{}$)
1. \( \tan ^{-1}(5.0) \)
2. \( \tan ^{-1}(0.5) \)
3. \( \tan ^{-1}(0.2) \)
4. \( \tan ^{-1}(2.0)\)