A jar '$$\mathrm{P}$$' is filled with gas having pressure, volume and temperature $$\mathrm{P}, \mathrm{V}, \mathrm{T}$$ respectively. Another gas jar $$Q$$ filled with a gas having pressure $$2 \mathrm{P}$$, volume $$\frac{\mathrm{V}}{4}$$ and temperature $$2 \mathrm{~T}$$. The ratio of the number of molecules in jar $$\mathrm{P}$$ to those in jar $$Q$$ is

The magnetic flux through a loop of resistance $$10 ~\Omega$$ varying according to the relation $$\phi=6 \mathrm{t}^2+7 \mathrm{t}+1$$, where $$\phi$$ is in milliweber, time is in second at time $$\mathrm{t}=1 \mathrm{~s}$$ the induced e.m.f. is

A thin rod of length '$$L$$' is bent in the form of a circle. Its mass is '$$M$$'. What force will act on mass '$$m$$' placed at the centre of this circle?

( $$\mathrm{G}=$$ constant of gravitation)

The coil of an a.c. generator has 100 turns, each of cross-sectional area $$2 \mathrm{~m}^2$$. It is rotating at constant angular speed $$30 ~\mathrm{rad} / \mathrm{s}$$, in a uniform magnetic field of $$2 \times 10^{-2} \mathrm{~T}$$. If the total resistance of the circuit is $$600 ~\Omega$$ then maximum power dissipated in the circuit is