A metallic rod of mass per unit length $$0.5 \mathrm{~kg} \mathrm{~m}^{-1}$$ is lying horizontally on a smooth inclined plane which makes an angle of $$30^{\circ}$$ with the horizontal. A magnetic field of strength $$0.25 \mathrm{~T}$$ is acting on it in the vertical direction. When a current $$I$$ is flowing through it, the rod is not allowed to slide down. The quantity of current required to keep the rod stationary is

A proton moves with a velocity of $$5 \times 10^6 \hat{\mathbf{\widehat j} m \mathrm{~m}^{-1}}$$ through the uniform electric field, $$\mathbf{\overrightarrow E}=4 \times 10^6[2 \hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+0.1 \hat{\mathbf{k}}] \mathrm{Vm}^{-1}$$ and the uniform magnetic field $$\mathbf{\overrightarrow B}=0.2[\hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+\hat{\mathbf{k}}] \mathrm{T}$$. The approximate net force acting on the proton is

A solenoid of length $$50 \mathrm{~cm}$$ having 100 turns carries a current of $$2.5 \mathrm{~A}$$. The magnetic field at one end of the solenoid is

A circular coil of wire of radius $$r$$ has $$n$$ turns and carries a current $$I$$. The magnetic induction $$B$$ at a point on the axis of the coil at a distance $$\sqrt{3} r$$ from its centre is