Question

Derive the charge-to-mass ratio formula for an electron in terms of ∆Vd (deflecting voltage), ∆Va (accelerating voltage), I, R, d, k, and N.

Answer 1

Note to student : If you want proper solution, please give full details of problem. You have not explained various terms such as R,k , d and N. So I had to assume that they mean Radius of circular path of electron, Geometry constant, distance between deviating plates and Number of turns of coli respectively.

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The energy gained by charge q accelerated through potential diffrence = q X = e X

where e = charge of one electron,   

This energy is gained by electron as kinetic energy =

= 2 x e x /m or ............... (1)

when beam of electrons enters B field it experiences a force = e x x B

when we a apply across plates and balance the magnetic force with electric force for nil deviation of electron beam

e B = e = where is electric field due to deviating voltage , d is distance between plates

So =

By using (1) we can write ,   

or e/m =

where B is replaced by N X I X k N= no of turns I = current in turn k= a constant depending on geometry of the coilused to create the B field.

Alternative solution

where R = Radius of circular path of Electron in Magnetic Field

combining this with equation (1) and eliminating we get

e/m =

Now B = k I N where k is a constant depending on Geometry of coil producing B and I is current and N=no of turns

e/m =