Question

Two coils have a mutual inductance \(0.005 \mathrm{H}\). The current changes in a coil according to equation \(\mathrm{I}=\mathrm{I}_{0}\) sin ot. Where \(I_{0}=10 \mathrm{~A} \& \omega=100 \pi \mathrm{rad} \mathrm{s}^{-1}\). The maximum value of emf in second coil is ............. (a) \(2 \pi\) (b) \(5 \pi\) (c) \(\pi\) (d) \(4 \pi\)

Short answer

The maximum value of emf in the second coil is \(5\pi\).

Step-by-step solution

We are given the mutual inductance M = 0.005 H, the maximum current \(I_0 = 10 A\), the angular frequency \(\omega = 100\pi rad/s\), and the equation of the current change in the first coil: \(I = I_0\sin(\omega t)\). #Step 2: Apply Faraday's law of electromagnetic induction#

According to Faraday's law of electromagnetic induction, the induced emf in the second coil is given by \(-\frac{d\Phi}{dt}\), where \(\Phi\) is the mutual magnetic flux. The mutual magnetic flux is the product of mutual inductance (M) and current (I) in the first coil, i.e., \(\Phi = MI = M I_0\sin(\omega t)\). #Step 3: Compute the time derivative of the mutual magnetic flux#

To find the induced emf, first compute the time derivative of the magnetic flux: \(\frac{d\Phi}{dt} = M I_0 \omega\cos(\omega t)\). #Step 4: Find the maximum value of induced emf in the second coil#

The maximum value of the induced emf can be found when the cosine term in the time derivative of the magnetic flux is at its maximum (\(\cos(\omega t) = 1\)). Therefore, the maximum induced emf is: \(\frac{d\Phi}{dt_{max}} = MI_0\omega\). #Step 5: Substitute the given values and calculate the maximum induced emf#

Now, substitute the given values of M, \(I_0\), and \(\omega\) in the equation: \(Max\:emf = 0.005\cdot 10\cdot 100\pi\). Calculate the result: \(Max\:emf = 5\pi\). #Step 6: Choose the correct option#

Based on our calculations, the maximum value of emf in the second coil is \(5\pi\). Therefore, the correct option is (b) \(5 \pi\).