A charged particle is moving in a uniform magnetic field in a circular path of radius ' $R$ '. When the kinetic energy of the particle is increased to three times, then the new radius will be

In the Bohr model of hydrogen atom, the centripetal force is furnished by the coulomb attraction between the proton and the electron. If ' $r_0$ ' is the radius of the ground state orbit, ' $m$ ' is the mass, ' e ' is the charge on the electron and ' $\varepsilon_0$ ' is the permittivity of vacuum, the speed of the electron is

A Carnot's cycle operating between $T_H=600 \mathrm{~K}$ and $T_c=300 \mathrm{~K}$ produces 1.5 kJ of mechanical work per cycle. The heat transferred to the engine by the reservoir is

Two concentric circular coils having radii ' $r_1{ }^{\prime}$ and ' $r_2$ ' $\left(r_2 \ll r_1\right)$ are placed co-axially with centres coinciding. The mutual inductance of the arrangement is ( $\mu_0=$ permeability of free space) (Both coils have single turn)