Two small spheres each of mass \(10 \,mg\) are suspended from a point by threads \(0.5 \,m\) long. They are equally charged and repel each other to a distance of \(0.20 \,m\). The charge on each of the sphere is \(\frac{ a }{21} \times 10^{-8} \, C\). The value of \(a\) will be ...... . \(\left[\right.\) Given \(\left.g=10 \,ms ^{-2}\right]\)

A body of mass \(2 \mathrm{~kg}\) begins to move under the action of a time dependent force given by \(\overrightarrow{\mathrm{F}}=\left(6 \mathrm{t} \hat{\mathrm{i}}+6 \mathrm{t}^2 \hat{\mathrm{j}}\right) \mathrm{N}\). The Power developed by the force at the time \(t\) is given by _______.

The velocity of a small ball of mass \(0.3\,g\) and density \(8\,g / cc\) when dropped in a container filled with glycerine becomes constant after some time. If the density of glycerine is \(1.3\,g / cc\), then the value of viscous force acting on the ball will be \(x \times 10^{-4}\,N\), the value of \(x\) is [use \(g=10\,m / s ^{2}\) ] 

Regarding self-inductance:
A. The self-inductance of the coil depends on its geometry.
B. Self-inductance does not depend on the permeability of the medium.
C. Self-induced e.m.f. opposes any change in the current in a circuit.
D. Self-inductance is electromagnetic analogue of mass in mechanics.
E. Work needs to be done against self-induced e.m.f. in establishing the current.
Choose the correct answer from the options given below:

A thin convex lens and a thin concave lens are kept in contact and are co-axial. Which of the following statements is correct for this combination of two lenses ?

A block of mass \(10\,kg\) is moving along \(x\)-axis under the action of force \(F=5 x\; N\). The work done by the force in moving the block from \(x =2\,m\) to \(4\,m\) will be ............\(J\).

A solid sphere of mass \(500\,g\) and radius \(5\,cm\) is rotated about one of its diameter with angular speed of \(10\,rad \, s ^{-1}\). If the moment of inertia of the sphere about its tangent is \(x \times 10^{-2}\) times its angular momentum about the diameter. Then the value of \(x\) will be ..............

If \(\varepsilon_0\) is the permittivity of free space and \(E\) is the electric field, then \(\varepsilon_0 E^2\) has the dimensions is _______.

Consider the \(LR\) circuit shown in the figure. If the switch \(S\) is closed at \(t = 0\) then the amount of charge that passes through the battery between \(t = 0\) and \(t = \frac{L}{R}\) is

A current \(i\) is flowing in a straight conductor of length \(L.\) The magnetic induction at a point on its axis at a distance \(\frac {L}{4}\) from its centre will be

An electric dipole of moment \(\overrightarrow{\mathrm{p}}=(-\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \times 10^{-29}\; \mathrm{C} \cdot \mathrm{m}\) is at the origin \((0,0,0) .\) The electric field due to this dipole at \(\overrightarrow{\mathrm{r}}=+\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}\) (note that \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{p}}=0)\) is parallel to

A bullet of mass \(50 \mathrm{~g}\) is fired with a speed \(100 \mathrm{~m} / \mathrm{s}\) on a plywood and emerges with \(40 \mathrm{~m} / \mathrm{s}\). The percentage loss of kinetic energy is _______.