Question

If the de Broglie wavelength of a proton is 70.0 fm, (a) what is the speed of the proton and (b) through what electric potential would the proton have to be accelerated to acquire this speed?

Answer 1

as given, de Broglie wavelength, = 70 fm = 70 * 10^-15 m

Mass of a proton, m = 1.67 * 10^-27 kg

charge carried by a proton, q = 1.6 * 10^-19 C

a)

Let v be the speed of the protonmomentum of proton, P = h /

here , h is Plank's constant = 6.63 * 10^-34 Js

also momemtum, P = m* v

so, m* v = h /

so, velocity, v = h / m = 6.63 * 10^-34 / (1.67 * 10^-27 * 70 * 10^-15) = 5.6715 * 10⁶ m/s

b)

let V be the electric potential required to accelerate the proton ,

Energy associated with a moving proton = 1/2 mv²

Energy required to move the proton with this speed = qV

apply law of conservation of energy :

qV = 1/2 *mv²

So, electric potential, V = 1/2 * mv²/ q

= 1/2 * 1.67 * 10^-27 * (5.6715 * 10⁶)² / (1.6 * 10^-19) = 167866.69 V = 167.87 kV