Question

An electron revolving in the \( n^{th} \) Bohr orbit has magnetic moment \( \mu \). If \( \mu_n \) is the value of \( \mu \), the value of \( x \) is:

• A. 2
• B. 1
• C. 3
• D. 0

Answer 1

The Correct Option is B

The magnetic moment \( \mu \) of an electron in the \( n^{th} \) Bohr orbit is given by the formula:

\[ \mu = \frac{e}{2m} r^2, \]

where:
- \( e \) is the charge of the electron,
- \( m \) is the mass of the electron,
- \( r \) is the radius of the orbit.

For a hydrogen atom, the radius of the \( n^{th} \) Bohr orbit is given by:

\[ r_n = n^2 \frac{h^2}{4\pi^2 ke^2m}, \]

where \( h \) is Planck’s constant and \( k \) is Coulomb’s constant.

Substituting \( r_n \) into the magnetic moment formula gives:

\[ \mu_n = \frac{e}{2m} \left( n^2 \frac{h^2}{4\pi^2 ke^2m} \right) = \frac{eh^2n^2}{8\pi^2 ke^2m^2}. \]

This simplifies to:

\[ \mu_n = n^2 \left( \frac{eh^2}{8\pi^2 ke^2m^2} \right). \]

Since \( \mu_1 \) (the magnetic moment for the first orbit) can be taken as a reference, we can find the ratio:

\[ \frac{\mu_n}{\mu_1} = n^2. \]

Thus, when \( n = 1 \):

\[ \frac{\mu_n}{\mu_1} = 1^2 = 1. \]

Therefore, the value of \( x \) is: 1