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.

An electron enters the space between the plates of a charged capacitor as shown. The charge density on the plate is $\sigma $. Electric intensity in the space between the plates is $E$. A uniform magnetic field $B$ also exists in that space perpendicular to the direction of $E$. The electron moves perpendicular to both $\vec E$ and $\vec B$ without any change in direction. The time taken by the electron to travel a distance $\ell $ is the space is

An electron is travelling along the $x$-direction. It encounters a magnetic field in the $y$-direction. Its subsequent motion will be

An electron is projected with velocity $\vec v$ in a uniform magnetic field $\vec B$ . The angle $\theta$  between $\vec v$ and $\vec B$  lines between $0^o$ and $\frac{\pi}{2}$ . It velocity $\vec v$ vector returns to its initial  value in time interval of 

A particle having some charge is projected in $x-y$ plane with a speed of $5\ m/s$ in a region having uniform magnetic field along $z-$ axis. Which of the following cannot be the possible value of velocity at any time ?