91影视

1. Turns of wire N = 100
2. Rectangular loop dimension 50.0 cm by 30.0 cm
3. Magnetic field B = 3.50 T
4. Coil rotation 1000 rev/min

From Faraday鈥檚 law, evaluate the induced emf from the number of turns and the change in the magnetic field lines that pass through the loop. Find the magnetic field lines with the given magnetic field and the area of the rectangular loop.

Faraday's law of electromagnetic inductionstates, Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in it.

Formulae are as follows:

$$\begin{gathered} \mathbf{蠒}=鈭�\mathrm{B}.\mathrm{dA} \\ \mathrm{蔚}=-\mathrm{N}\frac{\mathrm{d}\mathbf{蠒}}{\mathrm{dt}} \end{gathered}$$

Where,$\mathbf{蠒}$ is magnetic flux, B is magnetic field, A is area, 饾渶 is emf, Nis number of turns.

The coil rotation is,

$$\begin{gathered} 1000\mathrm{rev}/\min =\frac{1000\mathrm{rev}}{1\min }脳\frac{1\min }{60\mathrm{s}}脳\frac{2\mathrm{蟺}\mathrm{rad}}{1\mathrm{rev}}=104.7\mathrm{rad}/\sec \\ \mathrm{蝇}=104.7\mathrm{rad}/\sec \end{gathered}$$

The magnetic field lines $\mathbf{蠒}$with the given magnetic field and the area of the rectangular loop is given as,

$$\begin{gathered} \mathbf{蠒}=鈭�B.dA \\ \mathbf{蠒}=鈭�BdA\cos 胃 \end{gathered}$$

But$胃=蝇trad$

$$\begin{gathered} \mathbf{蠒}=鈭�BdA\cos \left(蝇t\right) \\ \mathbf{蠒}=BdA\cos \left(蝇t\right) \end{gathered}$$

By differentiating the magnetic field lines with respect to time,

$$\begin{gathered} \frac{\mathbf{d}\mathbf{蠒}}{\mathbf{dt}}=\frac{d}{dt}BA\cos 蝇t \\ \frac{\mathbf{d}\mathbf{蠒}}{\mathbf{dt}}=-蝇BA\cos \left(蝇t\right) \end{gathered}$$

From Faraday鈥檚 law, induced emf,

$$\begin{gathered} 蔚=-N\frac{\mathrm{d}\mathbf{蠒}}{\mathrm{dt}} \\ 蔚=-N\left(-蝇BA\sin \left(蝇t\right)\right)=N蝇BA\sin \\ 蔚=100脳104.7脳3.5脳\left(0.5脳0.3\right)\sin \left(蝇t\right) \end{gathered}$$

The induced emf will be maximum, when $\sin \left(蝇t\right)=1$.

$$\begin{gathered} 蔚=100脳\left(104.7\mathrm{rad}/\mathrm{s}\right)脳\left(3.5\mathrm{T}\right)脳\left(0.5m脳0.3m\right)脳1 \\ 蔚=5.496脳{10}^3鈮�5.5\mathrm{kV} \end{gathered}$$

Hence, the maximum value of emf is, $蔚=5.5\mathrm{kV}$.

Therefore, the induced emf in an electric generator can be determined using Faraday鈥檚 law when a coil of wire is wounded with the given number of turns.