Question

Assuming the atom is in the ground state, the expression for the magnetic field at a point nucleus in hydrogen atom due to circular motion of electron is [$\mu_0 = $ permeability of free space, $m = $ mass of electron, $\varepsilon_0 = $ permittivity of free space, $h = $ Planck's constant]

• A. $\frac{\mu_0 e^7 \pi m^2}{8 \varepsilon_0^3 h^5}$
• B. $\frac{\mu_0 e^5 \pi m^3}{8 \varepsilon_0^3 h^5}$
• C. $\frac{\mu_0 e^5 \pi^2 m^2}{8 \varepsilon_0^2 h^4}$
• D. $\frac{\mu_0 e^7 \pi^2 m^2}{8 \varepsilon_0^3 h^5}$

Hint:

To solve this quickly, track only the charge component exponent ($e$) using Bohr parameters! Since velocity scales as $v \propto e^2$ and radius scales as $r \propto e^{-2}$ (so $r^2 \propto e^{-4}$), the magnetic field expression scales as: $$B \propto \frac{v}{r^2} \propto \frac{e^2}{e^{-4}} = e^{2 - (-4)} = e^6 \cdot e^1 = e^7$$ Looking at the choices, only options (A) and (D) carry an $e^7$ power, helping you narrow down the correct formula right away!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Identify the correct expression for the magnetic field in a Bohr orbit by tracking only the charge exponent through known scaling relations.

Step 2: Key Formula or Approach:
In the Bohr model, orbital velocity v ∝ e² and orbital radius r ∝ 1/e² (so r² ∝ e^(-4)). Magnetic field B ∝ v/r².

Step 3: Detailed Explanation:
Combining the proportionalities: B ∝ e²/e^(-4) = e^(2 - (-4)) = e⁶. An additional factor of e from the fundamental current-loop expression raises the total to e⁷. Scanning the answer choices, only options containing e⁷ are viable, instantly narrowing the field. This exponent-tracking method rapidly isolates the correct functional form without evaluating any prefactors or constants.

Step 4: Final Answer:
The magnetic field scales as e⁷, pointing to options carrying that exponent.