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.

(N/A) The magnetic force on a charged particle is given by $\vec{F} = q(\vec{v} \times \vec{B})$. For the particle to travel undeflected,the magnetic force must be zero. This happens if the velocity $\vec{v}$ is parallel or anti-parallel to the magnetic field $\vec{B}$. Thus,the initial velocity of the particle is either parallel or anti-parallel to the magnetic field.
$(b)$ Yes,the final speed of the charged particle will be equal to its initial speed. The magnetic force acts perpendicular to the velocity vector at all times,meaning it does no work on the particle $(W = \vec{F} \cdot \vec{d} = 0)$. According to the work-energy theorem,the kinetic energy and hence the speed remains constant.
$(c)$ The electron travels from West to East. The electrostatic field is from North to South,so the electric force on the electron (which is negative) acts from South to North. To keep the electron undeflected,the magnetic force must act from North to South. Using Fleming's left-hand rule for a negative charge,the magnetic field must be applied in a vertically downward direction.