Before You Begin

Question #43789

Difficulty: 8/10 | 🟢 Free

The Question

A beam of free electrons with negligible kinetic energy recombines with ionized atoms into a specific energy state, emitting photons. The energy of this state is given by $E = -\frac{E_0}{(n^*)^2}$ , where $E_0$ is the ground state ionization energy and $n^*$ is the effective principal quantum number. Experimental measurements yield $E_0 = 54.0 \pm 0.54 \text{ eV}$ and $n^* = 3.00 \pm 0.03$ . The emitted photons are incident on a metallic surface with a work function $\Phi = 6.05 \text{ eV}$ . Using first-order error analysis for the maximum possible error, what is the maximum possible kinetic energy of the emitted photoelectrons?

Options

A.
$0.13 \text{ eV}$
B.
$0.07 \text{ eV}$
C.
$0.00 \text{ eV}$
D.
$0.01 \text{ eV}$

Check Your Answer

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

Energy Level Scaling|Work Function Dependence|Relative Error Propagation in Powers

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize quantum energy level transitions with standard error propagation, specifically targeting the power-rule in relative errors.

Students often treat the quantum number $n$ as an exact integer, but in effective models (like quantum defects), it carries measurement uncertainty.

The primary trap is failing to apply the chain rule correctly (missing the factor of 2 for $(n^*)^2$ in the denominator), which leads to an incorrect maximum photon energy of $6.12 \text{ eV}$ and a kinetic energy of $0.07 \text{ eV}$ .

Additionally, by setting the nominal photon energy ( $6.00 \text{ eV}$ ) below the work function ( $6.05 \text{ eV}$ ), students who ignore the error bounds entirely will incorrectly conclude that no photoelectric emission occurs ( $0.00 \text{ eV}$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Energy Level Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free