One electron gun is firing electrons with a speed V as shown in the figure. There is a target P which is to be hit. Electrons emerge from point 0 and line OP makes an angle with the direction of velocity of electrons. The length of OP is L.A uniform magnetic field B is applied parallel to line OP. Charge on the electron ise and the mass of electrons is m. What minimum magnetic field is needed so that electrons may hit the target P?

Explanation

Here, we can understand that electrons will follow a helical path with an axis parallel to line OP, so in every rotation the electron will touch some point on OP. Hence to strike the given target, length L must be an integral multiple of the pitch of the helix followed by electrons.
Pitch of the helix = Time period `xx` Component of
velocity parallel to field ` = (2pi m)/(eB) xx v cos theta`
Condition to strike the target
` L = n xx ` Pitch
`rArr " " L = n xx (2 pi m)/(eB) xx v cos theta`
`rArr " " B = n xx (2 pi mv cos theta)/(eL) `
For minimum possible magnetic field we can use `n = 1` .
`rArr " "B_("min") = (2 pi mv cos theta)/(eL)`