Two charges of same magnitude move in two circles of radii $R_1=R$ and $R_2=2 R$ in a region of constant uniform magnetic field $B _0$. The work $W_1$ and $W_2$ done by the magnetic field in the two cases respectively, are such that

Answer the following questions:

$(a)$ A magnetic field that varies in magnitude from point to point but has a constant direction (east to west) is set up in a chamber. A charged particle enters the chamber and travels undeflected along a straight path with constant speed. What can you say about the initial velocity of the particle?

$(b)$ A charged particle enters an environment of a strong and non-uniform magnetic field varying from point to point both in magnitude and direction, and comes out of it following a complicated trajectory. Would its final speed equal the initial speed if it suffered no collisions with the environment?

$(c)$ An electron travelling west to east enters a chamber having a uniform electrostatic field in north to south direction. Specify the direction in which a uniform magnetic field should be set up to prevent the electron from deflecting from its straight line path.

A proton of energy $200\, MeV$ enters the magnetic field of $5\, T$. If direction of field is from south to north and motion is upward, the force acting on it will be

Bob of a simple pendulum of length $l$ is made of iron . The pendulum is oscillating over a horizontal coil carrying direct current. If the time period of the pendulum is $T$ then

An electron and a positron are released from $(0, 0, 0)$ and $(0, 0, 1.5\, R)$ respectively, in a uniform magnetic field ${\rm{\vec B = }}{{\rm{B}}_0}{\rm{\hat i}}$ , each with an equal momentum of magnitude $P = eBR$. Under what conditions on the direction of momentum will the orbits be non-intersecting circles ?

A electron $(q = 1.6 \times 10^{-19}\, C)$ is moving at right angle to the uniform magnetic field $3.534 \times 10^{-5}\, T$. The time taken by the electron to complete a circular orbit is......$µs$