Part (i): Expression for Induced EMF in a Rotating Coil
Step 1 (Governing Law & Formula): [1 Mark]
The magnetic flux $\phi$ linked with a coil of $N$ turns, area $A$, in a magnetic field $B$ at an angle $\theta$ is given by:
$$\phi = N B A \cos(\theta)$$Since the coil rotates with angular speed $\omega$, $\theta = \omega t$. Thus, $\phi = N B A \cos(\omega t)$. According to Faraday’s Law of Induction, the induced emf $\varepsilon$ is:
$$\varepsilon = -\frac{d\phi}{dt}$$Step 2 (Mathematical Derivation): [1.5 Marks]
Substituting the expression for flux into Faraday's law:
$$\varepsilon = -\frac{d}{dt} (N B A \cos(\omega t))$$Since $N, B,$ and $A$ are constants:
$$\varepsilon = -N B A \frac{d}{dt} (\cos(\omega t))$$Using the derivative $\frac{d}{dt}(\cos(\omega t)) = -\omega \sin(\omega t)$:
$$\varepsilon = -N B A (-\omega \sin(\omega t))$$Step 3 (Final Expression): [0.5 Mark]
$$\varepsilon = N B A \omega \sin(\omega t)$$Where $\varepsilon_0 = N B A \omega$ is the peak value of the induced emf.
Part (ii): Calculation of Mutual Inductance
Step 1 (Formula & Given Data): [0.5 Mark]
Given: $r_1 = 1 \text{ cm} = 10^{-2} \text{ m}$, $r_2 = 100 \text{ cm} = 1 \text{ m}$.
The mutual inductance $M$ for two concentric coplanar loops (where $r_1 \ll r_2$) is derived from the flux through the smaller loop due to the magnetic field of the larger loop:
$$M = \frac{\mu_0 \pi r_1^2}{2 r_2}$$Step 2 (Mathematical Working): [1 Mark]
Substitute the values ($\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$ and $\pi^2 = 10$):
$$M = \frac{(4\pi \times 10^{-7}) \times \pi \times (10^{-2})^2}{2 \times 1}$$ $$M = \frac{4 \pi^2 \times 10^{-7} \times 10^{-4}}{2}$$ $$M = 2 \pi^2 \times 10^{-11}$$Step 3 (Final Answer): [0.5 Mark]
Using $\pi^2 = 10$:
$$M = 2 \times 10 \times 10^{-11} = 2 \times 10^{-10} \text{ H}$$💡 EL10 Concept Insight:
Think of the rotating coil like a spinning door in a breeze; as it turns, it "catches" different amounts of magnetic field lines, which forces electrons to move and creates electricity. For the loops, the big loop acts like a magnet, and the tiny loop inside just "listens" to the magnetic field created by