Six point charges are kept \( 60^{\circ} \) apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the centre of the circle is __________ .
(\( \epsilon \) is permittivity of free space)

The current flowing through an ac circuit is given by \(I=5 \sin (120 \pi t) A\) How long will the current take to reach the peak value starting from zero?

The diameter of a sphere is measured using a vernier caliper whose \(9\) divisions of main scale are equal to \(10\) divisions of vernier scale. The shortest division on the main scale is equal to \(1 \mathrm{~mm}\). The main scale reading is \(2 \mathrm{~cm}\) and second division of vernier scale coincides with a division on main scale. If mass of the sphere is \(8.635 \mathrm{~g}\), thedensity of the sphere \(1 \mathrm{~s}\) _______.

A positive ion \(A\) and a negative ion \(B\) has charges \(6.67 \times 10^{-19} \mathrm{C}\) and \(9.6 \times 10^{-10} \mathrm{C}\), and masses \(19.2 \times 10^{-27} \mathrm{~kg}\) and \(9 \times 10^{-27} \mathrm{~kg}\) respectively. At an instant, the ions are separated by a certain distance r. At that instant the ratio of the magnitudes of electrostatic force to gravitational force is \(P \times 10^{45}\), where the value of 10\(P\) is _______.
(Take \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-1}\) and universal gravitational constant as \(6.67 \times 10^{-11} \mathrm{Nm}^2 \mathrm{~kg}^{-2}\) )
Assume that charge may not be an integral multiple of electrons.

Expression for an electric field is given by \(\vec{E}=4000 x^2 \hat{i} \frac{V}{m}\). The electric flux through the cube of side \(20\,cm\) when placed in electric field (as shown in the figure) is \(.........V cm\).

The area of cross section of the rope used to lift a load by a crane is \(2.5 \times 10^{-4} m ^{2}\). The maximum lifting capacity of the crane is \(10\) metric tons. To increase the lifting capacity of the crane to \(25\) metric tons, the required area of cross section of the rope should be.\(.........\times 10^{-4} \,m ^{2}\) (take \(g =10\, ms ^{-2}\) )

The position of the image formed by the combination of lenses is _______. \(\mathrm{f}_1=10 \mathrm{~cm} \quad \mathrm{f}_2=10 \quad \mathrm{f}_3=30 \mathrm{~cm}\)

The acceleration due to gravity on the surface of earth is \(g\). If the diameter of earth reduces to half of its original value and mass remains constant, then acceleration due to gravity on the surface of earth would be :

An electron with kinetic energy \(5 \mathrm{eV}\) enters a region of uniform magnetic field of \(3 \mu \mathrm{T}\) perpendicular to its direction. An electric field \(\mathrm{E}\) is applied perpendicular to the direction of velocity and magnetic field. electron moves along the same path, The value of \(\mathrm{E}\) is _______ \(\mathrm{NC}^{-1}\). (Given, mass of electron \(=9 \times 10^{-31} \mathrm{~kg}\), electric charge \(=1.6 \times 10^{-19} \mathrm{C}\) )

The resistance of an electrical toaster has a temperature dependence given by \(R\left( T \right) = {R_0}\left[ {1 + \alpha \left( {T - {T_0}} \right)} \right]\) in its range of operation. At \({T_0} = 300\,K,R = 100\,\Omega \) and at \(T = 500\,K,\,R = 120\,\Omega \). The toaster is connected to a voltage source at \(200\, V\) and its temperature is raised at a constant rate from \(300\) to \(500\, K\) in \(30\, s\). The total work done in raising the temperature is