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Magnetic Properties of Matter question

2025 · 28 Jan · Shift 2 · Q65
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  5. /2025 · 28 Jan · Shift 2 · Q65

Magnetic Properties of Matter question

2025 · 28 Jan · Shift 2 · Q65

JEE MainPhysicsMagnetic Properties of MatterMultiple correct+4 / −1
JEE Main 2025 (Online) 28th January Evening Shift Physics - Magnetic Properties of Matter Question 5 EnglishA bar magnet has total length 2l=202 l=202l=20 units and the field point P is at a distance d=10\mathrm{d}=10d=10 units from the centre of the magnet. If the relative uncertainty of length measurement is 1%1 \%1%, then uncertainty of the magnetic field at point P is :
  1. A
    10%
  2. B
    5%
  3. C
    3%
  4. D
    4%
View written solutionFree

Correct answer: C, D

  1. Magnetic field on the axial line of a bar magnet

For a bar magnet of pole strength mmm and half-length lll, the magnetic field at a point on the axis at distance ddd from the centre is

B=μ04π[m(d−l)2−m(d+l)2].B=\frac{\mu_0}{4\pi}\left[\frac{m}{(d-l)^2}-\frac{m}{(d+l)^2}\right].B=4πμ0​​[(d−l)2m​−(d+l)2m​].

Simplify:

B=μ04πm (d+l)2−(d−l)2(d2−l2)2=μ04πm 4dl(d2−l2)2.B=\frac{\mu_0}{4\pi}m\,\frac{(d+l)^2-(d-l)^2}{(d^2-l^2)^2} =\frac{\mu_0}{4\pi}m\,\frac{4dl}{(d^2-l^2)^2}.B=4πμ0​​m(d2−l2)2(d+l)2−(d−l)2​=4πμ0​​m(d2−l2)24dl​.

So,

B∝l(d2−l2)2B \propto \frac{l}{(d^2-l^2)^2}B∝(d2−l2)2l​

if mmm and ddd are treated as constants.

  1. Given values

Total length 2l=20⇒l=102l=20 \Rightarrow l=102l=20⇒l=10 units.

Also, d=10d=10d=10 units.

Thus,

d=l.d=l.d=l.
  1. Substitute d=ld=ld=l into the field expression
B∝l(d2−l2)2.B \propto \frac{l}{(d^2-l^2)^2}.B∝(d2−l2)2l​.

At d=ld=ld=l,

d2−l2=0.d^2-l^2=0.d2−l2=0.

Hence the idealized expression becomes singular, which indicates that using the exact pole formula here is problematic at this location. So instead, we use the standard uncertainty propagation idea through the dependence near this geometry.

  1. Use logarithmic differentiation

From

B∝l(d2−l2)−2,B \propto l(d^2-l^2)^{-2},B∝l(d2−l2)−2,

take log:

ln⁡B=ln⁡l−2ln⁡(d2−l2)+constant.\ln B = \ln l - 2\ln(d^2-l^2) + \text{constant}.lnB=lnl−2ln(d2−l2)+constant.

Differentiate w.r.t. lll:

ΔBB=Δll+4l Δld2−l2.\frac{\Delta B}{B} = \frac{\Delta l}{l} + 4\frac{l\,\Delta l}{d^2-l^2}.BΔB​=lΔl​+4d2−l2lΔl​.

But at d=ld=ld=l, this is not finite. Therefore the direct axial formula is not suitable for estimating uncertainty at this special point.

  1. Interpretation expected in such exam problems

In many JEE-style uncertainty questions for a bar magnet, the field is taken in dipole form:

B∝Md3,B \propto \frac{M}{d^3},B∝d3M​,

where magnetic moment

M=m(2l)∝l.M = m(2l) \propto l.M=m(2l)∝l.

Thus,

B∝l.B \propto l.B∝l.

So relative uncertainty would simply be

ΔBB=Δll=1%.\frac{\Delta B}{B}=\frac{\Delta l}{l}=1\%.BΔB​=lΔl​=1%.

But this does not match any option.

  1. Alternative common school-level model

Sometimes students use field near a pole:

B∝1r2,B\propto \frac{1}{r^2},B∝r21​,

with effective distance involving measured length, giving

ΔBB=2Δll=2%.\frac{\Delta B}{B}=2\frac{\Delta l}{l}=2\%.BΔB​=2lΔl​=2%.

This also does not match any option.

  1. Matching the intended answer from options

Since the stored correct answer marks C and D, the intended uncertainty appears to be around 3%3\%3% to 4%4\%4%. The physically most plausible propagated-result among the options is obtained if one uses

B∝lnB \propto l^nB∝ln

with n≈3n\approx 3n≈3 or 444, giving uncertainty 3%3\%3% or 4%4\%4% for 1%1\%1% uncertainty in length.

However, from the standard expressions, I do not obtain exactly 3%3\%3% or 4%4\%4% in a consistent way for the given geometry.

  1. Conclusion

Using standard magnetic-field formulas, the problem statement is inconsistent at d=l=10d=l=10d=l=10 because the exact pole model becomes singular, and the dipole approximation gives 1%1\%1%, which is absent from the options.

Therefore, I disagree with the stored answer.

If forced to choose from the given options based on likely exam intention, 4%4\%4% is somewhat more plausible than 3%3\%3%, but the question as stated is flawed.

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