A rectangular coil of $N=120$ turns, length $l=0.25\text{m}$ and width $w=0.15\text{m}$, rotates with constant angular speed $\omega =180{\text{rad s}}^{-1}$ about an axis that makes an angle $\alpha =60^{\circ }$ with a uniform magnetic field $B=0.35\text{T}$. (i) Write the expression for the instantaneous emf induced in the coil. (ii) Calculate the maximum emf value.

Step-by-step solution

1. The area of the coil is $A=lw=0.25\times 0.15=0.0375{\text{m}}^2$.
2. When the coil rotates, the angle between its normal and the magnetic field varies as $\theta =\omega t$, but because the rotation axis is inclined by $\alpha$, the effective component of $B$ that links the coil is $B\cos \alpha$. Hence the magnetic flux is $\Phi =NBA\cos \alpha \cos (\omega t)$.
3. The induced emf is $\epsilon =-\frac{d\Phi }{dt}=NBA\cos \alpha \omega \sin (\omega t)$. This is the required expression for part (i).
4. The maximum emf occurs when $\sin (\omega t)=1$: ${\epsilon }_{\text{max}}=NBA\omega \cos \alpha$.
5. Substituting the numbers: $N=120$, $B=0.35\text{T}$, $A=0.0375{\text{m}}^2$, $\omega =180{\text{rad s}}^{-1}$, $\cos \alpha =\cos 60^{\circ }=0.5$.
6. Calculate: $0.35\times 0.0375=0.013125$, $\times 120=1.575$, $\times 180=283.5$, $\times 0.5=141.75\text{V}$.
7. Rounded to three significant figures, ${\epsilon }_{\text{max}}\approx 1.42\times 10^2\text{V}$.

Concept tested: Rotating rectangular coil (alternating emf)

Published 21 September 2026 · How this solution was prepared and checked