Question

The area of a plane coil 1000 turns is 500 cm\(^2\) and it is held perpendicular to a uniform magnetic field of \(4 \times 10^{-4}\) weber/m\(^2\). It is turned through 180° angle in 0.1 second. Calculate the average induced e.m.f. produced in the coil.

Hint:

The induced e.m.f. depends on the rate of change of the magnetic flux. A larger change in flux or faster rotation results in a higher induced e.m.f.

Answer 1

The average induced e.m.f. in a coil can be calculated using Faraday's law of electromagnetic induction, which states that the induced e.m.f. is equal to the rate of change of magnetic flux through the coil. The formula for induced e.m.f. is:
\[ \varepsilon = - N \frac{\Delta \Phi}{\Delta t} \] Where: - \(N\) is the number of turns of the coil, - \(\Delta \Phi\) is the change in magnetic flux, - \(\Delta t\) is the time interval. The magnetic flux \(\Phi\) is given by: \[ \Phi = B \times A \times \cos \theta \] Where: - \(B\) is the magnetic field strength, - \(A\) is the area of the coil, - \(\theta\) is the angle between the normal to the plane of the coil and the magnetic field. Step 1: Calculate the initial and final magnetic flux.
Initially, the coil is perpendicular to the magnetic field, so \(\theta = 0^\circ\), and \(\cos 0^\circ = 1\). The initial magnetic flux is:
\[ \Phi_{\text{initial}} = B \times A \times 1 = (4 \times 10^{-4}) \times (500 \times 10^{-4}) = 2 \times 10^{-2} \, \text{Wb} \] Finally, after the coil is turned through \(180^\circ\), \(\theta = 180^\circ\), and \(\cos 180^\circ = -1\). The final magnetic flux is:
\[ \Phi_{\text{final}} = B \times A \times (-1) = - (2 \times 10^{-2}) = - 2 \times 10^{-2} \, \text{Wb} \] Step 2: Calculate the change in magnetic flux.
The change in magnetic flux is:
\[ \Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} = - 2 \times 10^{-2} - 2 \times 10^{-2} = - 4 \times 10^{-2} \, \text{Wb} \] Step 3: Calculate the induced e.m.f.
The time interval \(\Delta t = 0.1 \, \text{s}\), and the number of turns \(N = 1000\). Now, using Faraday's law:
\[ \varepsilon = - 1000 \times \frac{- 4 \times 10^{-2}}{0.1} = 400 \, \text{V} \] Thus, the average induced e.m.f. is \(400 \, \text{V}\).