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Question #43796

Difficulty: 8/10 | 🟢 Free

The Question

In an experiment to determine the Rydberg constant $R$ , a beam of $H_\alpha$ photons (corresponding to the transition from $n=3$ to $n=2$ in hydrogen) is passed through a uniform transverse magnetic field of $B = 2.0 \text{ T}$ and falls normally on a transmission diffraction grating. The first-order ( $m=1$ ) principal maximum is observed at a diffraction angle $\theta = 30^\circ$ . The angular scale of the spectrometer has a least count of $\Delta \theta = \frac{1}{500\sqrt{3}} \text{ rad}$ . Assuming the grating constant $d$ is known exactly, what is the maximum percentage error in the calculated value of the Rydberg constant $R$ ?

Options

A.
$\frac{1}{5}\%$
B.
$\frac{1}{15}\%$
C.
$\frac{36}{25}\%$
D.
$\frac{1}{5\sqrt{3}}\%$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Rydberg Constant Sensitivity|Photon Neutrality|Least Count and Resolution Limits

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to treat the Rydberg constant not as a fixed fundamental constant, but as a derived experimental quantity subject to error propagation.

The primary trap is twofold: First, students may incorrectly apply a Lorentz force deflection to the photons due to the magnetic field, failing to recognize photon neutrality.

Second, a common mathematical trap occurs during error propagation; students often incorrectly differentiate the grating equation (yielding a $\tan \theta$ dependence instead of $\cot \theta$ ) or improperly scale the error by the quantum number factor ( $36/5$ ), leading to incorrect percentage errors.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Rydberg Constant Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free