Question

An electron enters a uniform magnetic field of flux density \(1.2\ \text{Wb/m}^2\). The energy difference (in eV) between electrons having spins parallel and anti-parallel to the field is:
(Given: \(\mu_B = 9.3 \times 10^{-24}\ \text{J/T}\))

• A. \(3.95 \times 10^{-5}\ \text{eV}\)
• B. \(13.95 \times 10^{-5}\ \text{eV}\)
• C. \(23.95 \times 10^{-5}\ \text{eV}\)
• D. \(33.95 \times 10^{-5}\ \text{eV}\)

Hint:

Energy gap between the two spin orientations is \(2\mu_B B\); then convert joules to eV.

Answer 1

The Correct Option is B

Step 1: The magnetic potential energy of an electron spin magnetic moment in a field \(B\) is \(U = \pm \mu_B B\), one sign for spin parallel and the other for spin anti-parallel to the field.

Step 2: The energy separation between the two orientations is
\[\Delta U = 2\mu_B B\]
Step 3: Substitute \(\mu_B = 9.3 \times 10^{-24}\ \text{J/T}\) and \(B = 1.2\ \text{T}\):
\[\Delta U = 2 \times 9.3 \times 10^{-24} \times 1.2 = 22.32 \times 10^{-24}\ \text{J} = 2.232 \times 10^{-23}\ \text{J}\]
Step 4: Convert to eV by dividing by \(1.6 \times 10^{-19}\ \text{J/eV}\):
\[\Delta U = \frac{2.232 \times 10^{-23}}{1.6 \times 10^{-19}} = 1.395 \times 10^{-4}\ \text{eV} = 13.95 \times 10^{-5}\ \text{eV}\]
This matches option (B).
\[\boxed{\Delta U = 13.95 \times 10^{-5}\ \text{eV}}\]