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Question #43226

Difficulty: 8/10 | 🟢 Free

The Question

A modified optical system has a magnification $M$ given by the formula: $M = \frac{k L \sin \theta}{(L - f_e)^3}$ where $k$ is an exact constant with dimensions of $\text{length}^2$ , $L$ is the measured tube length, $f_e$ is the exact focal length of the eyepiece, and $\theta$ is the measured small angular aperture of the objective lens.

In a particular experiment, the measured values are $L = (15.00 \pm 0.15) \text{ cm}$ and $\theta = (0.020 \pm 0.001) \text{ rad}$ . The focal length of the eyepiece is exactly $f_e = 10.00 \text{ cm}$ .

Assuming the small angle approximation $\sin \theta \approx \theta$ is valid, what is the maximum percentage error in the calculated value of the magnification $M$ ?

Options

A.
13.0%
B.
15.0%
C.
9.0%
D.
7.0%

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Microscope Tube Length|Dimensional Consistency and Transcendental Functions|Relative Error Propagation in Powers

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly propagate errors when a single measured physical quantity (tube length $L$ ) appears in multiple terms of an equation.

A common misconception is to treat the $L$ in the numerator and the $(L - f_e)$ in the denominator as independent variables, leading to a simple addition of their individual relative errors ( $\frac{\Delta L}{L} + 3\frac{\Delta L}{L-f_e}$ ).

This ignores the dependent variable correlation and yields an incorrect higher error of 15.0%.

The correct approach requires logarithmic differentiation and algebraic grouping of the $dL$ terms before applying the absolute value for maximum error, which correctly accounts for how variations in $L$ simultaneously affect both the numerator and denominator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Microscope Tube Length
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free