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) \(5 \pi\).

Key concepts

Faraday's law of electromagnetic induction

Faraday's law of electromagnetic induction is a fundamental principle describing how a change in magnetic flux can induce an electromotive force (emf) in a coil. This principle is foundational in understanding how transformers, electric generators, and inductors operate.
Faraday discovered that a voltage is induced in a conductor when it is exposed to a changing magnetic field. The induced voltage can be expressed mathematically as \(-\frac{d\Phi}{dt}\), where \(\Phi\) represents the magnetic flux through the coil. The negative sign in Faraday's law is a reflection of Lenz's law, indicating that the induced emf will always oppose the change in flux producing it.
If we consider two coils with mutual inductance, changes in the current of the first coil will induce a magnetic flux change in the second coil. As a result, an emf will be induced in the second coil according to Faraday's law. Understanding this principle allows us to calculate the amount of induced emf from the rate of change of magnetic flux.

Time derivative

The time derivative of a function represents how that function changes with respect to time. In the context of electromagnetic induction, the time derivative of the magnetic flux is crucial for determining the induced emf in a coil.
For a current changing according to \(I = I_0 \sin(\omega t)\), the magnetic flux \(\Phi = MI = MI_0 \sin(\omega t)\) through the coil depends on time. To find how quickly this flux changes, we compute its time derivative \(\frac{d\Phi}{dt}\).
By applying the chain rule, the derivative of \(MI_0 \sin(\omega t)\) becomes \(MI_0 \omega \cos(\omega t)\). This derivative tells us the rate at which the magnetic environment around the second coil is changing, directly yielding the induced emf due to that change. The importance of the derivative lies in its ability to quantify the change per unit of time, thus determining the magnitude and behavior of the induced emf.

Maximum induced emf

The maximum induced emf in a coil is observed when the change in magnetic flux is at its fastest. For sinusoidal functions like \(I = I_0 \sin(\omega t)\), the maximum occurs when the cosine function \(\cos(\omega t)\) achieves its highest value of 1.

• The maximum induced emf is calculated by considering the amplitude of the voltage produced without any reduction due to the sinusoidal variation.
• This occurs because the rate of change of magnetic flux is directly proportional to this cosine term. When \(\cos(\omega t) = 1\), the derivative \(\frac{d\Phi}{dt}\) reaches its peak value.

To find the maximum induced emf, substitute this peak condition into the derived expression for \(\frac{d\Phi}{dt}\).
Here, the expression simplifies to \(MI_0 \omega\). By plugging in the specific values given, such as mutual inductance \(M\), current amplitude \(I_0\), and angular frequency \(\omega\), you can find the precise maximum value of emf that the second coil experiences.