A uniform magnetic field (B) is applied between the plates of a capacitor in the outward direction as shown in the figure. Surface charge density on the capacitor plates is `sigma`.

An electron is fired in between the plates, parallel to them, as shown in the figure. The electron is found to move straight in-between the plates of capacitor. Neglect the gravity. If L is the length of the plates then caleulate the time taken by the electron to cross the plates.

Explanation

The electron is found moving straight and it is possible only when electric force is balanced by the magnetic force acting on the electron. We can see in the figure that the magnetic and electric forces acting on the electron are opposite to each other.
Let u be the velocity of projection of electron then for an undeflected motion, we can write the following equation:
` eE = evB`
` rArr" " v = E//B`
Electric field in-between the plates of capacitor can be written an `E = sigma//epsi_(0) B`. Hence, the velocity of electron becomes
`v = sigma//epsi_(0)B`.
We know that electron is moving straight with a uniform speed. So time taken by the electron to cross the region between the plates can be written as follows:
`t = L//v = epsi_(0) LB//sigma`.