A current-carrying coil has a magnetic moment \( \vec{m}=(3\hat{\imath}+4\hat{\jmath})\times10^{-2}\,\mathrm{A\,m^{2}} \). In which of the following uniform magnetic fields is the torque on the coil zero?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \vec{B}=(6\hat{\imath}+8\hat{\jmath})\,\mathrm{T} \)
A current loop of magnetic moment \( \vec{m} \) placed in a uniform magnetic field experiences the torque \( \vec{\tau}=\vec{m}\times\vec{B} \). This torque is zero when \( \vec{m} \) is parallel or antiparallel to \( \vec{B} \).
The components of \( \vec{m} \) are in the ratio 3 : 4, so a field along \( \vec{m} \) must have its components in the same ratio. The field \( (6\hat{\imath}+8\hat{\jmath})\,\mathrm{T} \) is twice \( (3\hat{\imath}+4\hat{\jmath}) \) and points along \( \vec{m} \). For this field:
\( \begin{aligned} \vec{\tau} &= \vec{m}\times\vec{B} \\ &= [(3\hat{\imath}+4\hat{\jmath})\times10^{-2}]\times(6\hat{\imath}+8\hat{\jmath}) \\ &= (3\times8-4\times6)\times10^{-2}\,\hat{k}\,\mathrm{N\,m} \\ &= 0 \end{aligned} \)
Hence, the coil feels no torque in the field \( \vec{B}=(6\hat{\imath}+8\hat{\jmath})\,\mathrm{T} \).
NCERT (2026-27):
Torque on a Rectangular Current Loop in a Uniform Magnetic Field – “… the torque τ vanishes when m is either parallel or antiparallel to the magnetic field B. This indicates a state of equilibrium as there is no torque on the coil (this also applies to any object with a magnetic moment m).”