A metal target with atomic number $Z=46$ is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio $r$ of the wavelengths of the $K_{\alpha}$-line and the cut-off is found to be $r=2$. If the same electron beam bombards another metal target with $Z=41$, the value of $r$ will be

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Modern Physics

A metal target with atomic number

Z = 46

is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio

r

of the wavelengths of the

K_{α}

-line and the cut-off is found to be

r = 2

. If the same electron beam bombards another metal target with

Z = 41

, the value of

r

will be

Answer: (A)

The ratio

r = \frac{λ _{K α}}{λ _{0}}

depends on the atomic number. For characteristic X-rays, the

K_{α}

wavelength is inversely proportional to

(Z - 1)^{2}

(Moseley's law).

The ratio scales as:

\frac{r _{2}}{r _{1}} = \frac{( Z _{1} - 1 ) ^{2}}{( Z _{2} - 1 ) ^{2}} = \frac{( 46 - 1 ) ^{2}}{( 41 - 1 ) ^{2}} = \frac{4 5 ^{2}}{4 0 ^{2}} = \frac{2025}{1600} = 1.265625

Therefore:

r_{2} = 1.265625 \times 2 = 2.53

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Single correct option