Two IIT-JEE 2007 Questions on X-rays

The following multiple choice question appeared in IIT-JEE 2007 question paper:

Electrons with de Broglie wave length λ fall on the target in an X-ray tube. The cut off wave length of the emitted X-rays is

(a) λ₀ = 2mcλ²/h (b) λ₀ = 2h/mc (c) λ₀ = 2m²c² λ³/h² (d) λ₀ = λ

X-rays of cut off (minimum) wave length for a given target voltage are obtained when the entire kinetic energy of the incident electron is converted into X-ray photon energy. The kinetic energy of the electron is p²/2m where ‘p’ is its momentum and ‘m’ is its mass.

But p = h/λ where ‘h’ is Planck’s constant [de Broglie relation] so that we have

h²/2mλ² = hc/λ₀

[The term on the right hand side is the energy of the X-ray photon of cut off wave length λ₀].

From the above, λ₀ = 2mcλ²/h

Here is an assertion-reason type multiple choice question which appeared in IIT-JEE 2007 question paper:

STATEMENT-1

If the accelerating potential in an X-ray tube is increased, the wave lengths of the characteristic X-rays do not change

because

STATEMENT-2

When an electron beam strikes the target in an x-ray tube, part of the kinetic energy is converted in to X-ray energy.

(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct

explanation for Statement-1

(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct

explanation for Statement-1

(c) Statement-1 is True, Statement-2 is False

(d) Statement-1 is False, Statement-2 is True

Wave lengths of characteristic X-rays are characteristic of the target material (since they depend only on the energy levels of the atoms in the target material). Therefore, their wave lengths do not change when the accelerating voltage is increased. Statement-1 is therefore true.

Statement-2 is true. But, it is not a correct explanation for statement-1. Characteristic X-rays originate because of electron transitions between discrete energy levels of the atoms in the target. Statement-2 is only a general statement regarding the process of X-ray production.