6. Induced EMF in Spiral Coil
Step 1: Flux Calculation
The coil has $N$ turns distributed from radius $0$ to $R$. The radius of the $n$-th turn is $r = \frac{n}{N}R$.
The magnetic field $B = B_0 \cos(\omega t)$ covers the entire coil area. The flux through a single turn of radius $r$ is $\phi = \pi r^2 B$.
Total flux $\Phi$ is the sum (or integral) over all turns:
$$ \Phi = \int_{0}^{N} \pi \left( \frac{n}{N} R \right)^2 B \, dn = \frac{\pi R^2 B}{N^2} \int_{0}^{N} n^2 \, dn $$ $$ \Phi = \frac{\pi R^2 B}{N^2} \left[ \frac{n^3}{3} \right]_0^N = \frac{\pi R^2 B}{N^2} \frac{N^3}{3} = \frac{1}{3} N \pi R^2 B $$Step 2: Induced EMF
Using Faraday’s Law:
$$ \mathcal{E} = -\frac{d\Phi}{dt} = -\frac{1}{3} N \pi R^2 \frac{dB}{dt} $$ $$ \frac{dB}{dt} = \frac{d}{dt} (B_0 \cos \omega t) = -B_0 \omega \sin \omega t $$ $$ \mathcal{E} = -\frac{1}{3} N \pi R^2 (-B_0 \omega \sin \omega t) $$
$$ \mathcal{E} = \frac{1}{3} \pi \omega N R^2 B_0 \sin(\omega t) $$