Derivation of the e/m Equation (a) Since the magnetic force always acts inward t

Question

Derivation of the e/m Equation

(a) Since the magnetic force always acts inward toward the center of the circular path taken by an electron in a magnetic field, find the equation of e/m in terms of v (speed) , r, and B by setting the magnetic force equal to the mass times the centripetal acceleration and solving the resultant equation for e/m.

(b) What is the equation for the kinetic energy of a mass m moving at a speed v?

(c) The electrons in an e/m tube are accelerated to a speed v by falling through a potential difference of V before being shot into the magnetic field. What is the equation for the potential energy lost by a charge e falling through a potential difference V?

(d) Set the potential energy lost by the electron equal to the kinetic energy gained by it as a result of its acceleration. Then use the equation you derived in part (a) to show that e/m is given by the expression: e/m = 2V/r^2 B^2

Explanation / Answer

a)

Here , for

centripetal force = magnetic force

mv^2/r = B * v * e

e/m = v/(B * r)

e/m is v/(B * r)

b)

kinetic energy = 0.5 * m * v^2

c)

Potential energy lost by electron = V * e

Potential energy lost by electron = e*V

d)

Now , as e * V = 0.5 * m * v^2

e/m = 2 /(V * v^2)

using equation from 1

e/m = 2/V *((e/m) * B * r)^2

e/m = 2V/r^2 B^2

hence , proved

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