To find the maximum value of the electromotive force (emf) induced in the second coil due to the changing current in the first coil, we can use the concept of mutual inductance. Let's break down the problem step by step.
Understanding Mutual Inductance
Mutual inductance occurs when a change in current in one coil induces an emf in another nearby coil. The relationship between the changing current and the induced emf can be described by the formula:
emf = -M (dI/dt)
Where:
- emf is the induced electromotive force in the second coil.
- M is the mutual inductance between the two coils (in henries).
- dI/dt is the rate of change of current in the first coil.
Given Values
From the problem, we have:
- Mutual inductance, M = 0.005 H
- Current in the first coil, I = X sin(wt), where X = 10 A and w = 100π rad/s.
Finding dI/dt
To find the rate of change of current, we first need to differentiate the current function with respect to time:
I(t) = 10 sin(100πt)
Now, let's differentiate:
dI/dt = 10 * 100π cos(100πt)
This simplifies to:
dI/dt = 1000π cos(100πt)
Calculating the Maximum Value of emf
The maximum value of the cosine function is 1. Therefore, the maximum rate of change of current, (dI/dt)_{max}, is:
(dI/dt)_{max} = 1000π
Now, we can substitute this value into the emf formula:
emf = -M (dI/dt)
Substituting the values we have:
emf = -0.005 H * 1000π
This results in:
emf = -5π V
Final Answer
The negative sign indicates the direction of the induced emf according to Lenz's law, but since we are interested in the magnitude, the maximum value of the induced emf in the second coil is:
5π V
Thus, the maximum value of emf in the second coil is indeed 5π volts. This process illustrates how mutual inductance allows us to calculate induced voltages based on changing currents in nearby coils.
