Moment of inertia of a cylinder of mass M, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is
I = $$M\left( {{{{R^2}} \over 4} + {{{L^2}} \over {12}}} \right)$$. If such a cylinder is to be made for a given mass of a material, the ratio $${L \over R}$$ for it to have minimum possible I is

The magnetic field of a plane electromagnetic wave is
$$\overrightarrow B = 3 \times {10^{ - 8}}\sin \left[ {200\pi \left( {y + ct} \right)} \right]\widehat i$$ T
where c = 3 $$ \times $$ 10⁸ ms^–1 is the speed of light. The corresponding electric field is :

A charged particle carrying charge 1 $$\mu $$C is moving
with velocity $$\left( {2\widehat i + 3\widehat j + 4\widehat k} \right)$$ ms^–1. If an external
magnetic field of $$\left( {5\widehat i + 3\widehat j - 6\widehat k} \right)$$× 10^–3 T exists in the region where the particle is moving then the
force on the particle is $$\overrightarrow F $$ × 10^–9 N. The vector $$\overrightarrow F $$ is :

A 750 Hz, 20 V (rms) source is connected to a resistance of 100 $$\Omega $$, an inductance of 0.1803 H and a capacitance of 10 $$\mu $$F all in series. The time in which the resistance (heat capacity 2 J/^oC) will get heated by 10^oC. (assume no loss of heat to the surroudnings) is close to :