Source: High school physics (Chinese)
Problem Sets:
Problem
A rectangular coil $abcd$ of area $S = 0.05 \text{ m}^2$ is placed in a uniform magnetic field of magnitude $B = 0.06 \text{ T}$. In the initial position the coil plane is perpendicular to $\vec{B}$. The coil can rotate about the axis $OO'$ which lies in the coil plane.
- When the coil plane is perpendicular to the magnetic field direction, what is the magnetic flux through the coil?
- When the coil is rotated by $60°$ about the $OO'$ axis from the position above, what is the magnetic flux through the coil?
(1) $\Phi_1 = BS = 3 \times 10^{-3} \text{ Wb}$
(2) $\Phi_2 = BS\cos 60° = 1.5 \times 10^{-3} \text{ Wb}$
Magnetic flux: $\Phi = BS\cos\theta$, where $\theta$ is the angle between $\vec{B}$ and the coil normal $\hat{n}$.
(1) Coil plane $\perp \vec{B}$ means $\hat{n} \parallel \vec{B}$, so $\theta = 0°$:
$$\Phi_1 = BS\cos 0° = BS = 0.06 \times 0.05 = 3 \times 10^{-3} \text{ Wb}$$(2) Rotating the coil by $60°$ about $OO'$ tilts $\hat{n}$ away from $\vec{B}$ by $60°$, so $\theta = 60°$:
$$\Phi_2 = BS\cos 60° = (3 \times 10^{-3}) \times \tfrac{1}{2} = 1.5 \times 10^{-3} \text{ Wb}$$