$$\mathop {\lim }\limits_{x \to 0} {{x\tan 2x - 2x\tan x} \over {{{\left( {1 - \cos 2x} \right)}^2}}}$$ equals :

A solid ball of radius R has a charge density $$\rho $$
given by $$\rho $$ = $$\rho $$_o (1 $$-$$ $${\raise0.5ex\hbox{$\scriptstyle r$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle R$}}$$) for 0 $$ \le $$ r $$ \le $$ R. The electric field outside the ball is :

A plane polarized monochromatic EM wave is traveling in vacuum along z direction such that at t = t₁ it is found that the electric field is zero at a spatial point z₁. The next zero that occurs in its neighbourhood is at z₂. The frequency of the electroagnetic wave is :

At the center of a fixed large circular coil of radius R, a much smaller circular coil of radius r is placed. The two coils are concentric and are in the same plane. The larger coil carries a current I. The smaller coil is set to rotate with a constant angular velocity $$\omega $$ about an axis along their common diameter. Calculate the emf induced in their smaller coil after a time t of its start of rotation.