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Question #42954

Difficulty: 8/10 | 🟢 Free

The Question

A bar magnet has a magnetic length $2a$ and a constant magnetic moment $m$ . The magnetic field magnitudes at an axial point and an equatorial point, both at a distance $r$ from the center of the magnet, are $B_{\text{axial}}$ and $B_{\text{eq}}$ respectively. A student evaluates the ratio $K = \frac{B_{\text{axial}}}{B_{\text{eq}}}$ for a specific distance $r = 2a$ . If the maximum percentage errors in the measurement of $r$ and $a$ are both $1.5\%$ , what is the maximum percentage error in the calculated value of $K$ ?

Options

A.
5.8%
B.
7.3%
C.
4.35%
D.
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

Magnetic Field on Axial vs Equatorial Line|Significant Figures in Mixed Operations|Relative Error Propagation in Powers

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of error propagation on a complex rational power function, specifically avoiding the common trap of using the short dipole approximation ( $r \gg a$ ).

If a student incorrectly assumes $r \gg a$ , they get a constant ratio $K=2$ , leading to a $0\%$ error.

Furthermore, it tests the chain rule in logarithmic differentiation for terms like $\ln(r^2 \pm a^2)$ .

Students often fail to take the absolute values of the coefficients to find the maximum possible error, which would incorrectly lead to a cancellation of errors.

Finally, it requires correctly relating the absolute differential $dr/a$ back to the given percentage error $dr/r$ using the substitution $r=2a$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Magnetic Field on Axial vs Equatorial Line
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free