To determine the time at which the electromotive force (emf) across a coil becomes zero, we first need to understand the relationship between current, inductance, and emf. The emf induced in an inductor is given by Faraday's law of electromagnetic induction, which states that the emf (ε) is equal to the negative rate of change of current (I) through the inductor. Mathematically, this can be expressed as:

Understanding the Induced EMF

The formula for the induced emf in an inductor is:

ε = -L (dI/dt)

Where:

• ε is the induced emf.
• L is the inductance of the coil (in henries).
• dI/dt is the derivative of the current with respect to time.

Given Values

In this case, we have:

• L = 2 mH = 2 x 10^-3 H
• I(t) = t^2 e^(-t)

Finding the Derivative of Current

Next, we need to find the derivative of the current function I(t) with respect to time t:

I(t) = t^2 e^(-t)

Using the product rule for differentiation, we can find dI/dt:

dI/dt = e^(-t) * (2t) + t^2 * (-e^(-t))

This simplifies to:

dI/dt = e^(-t) (2t - t^2)

Setting the EMF to Zero

Now, we can substitute dI/dt back into the emf equation:

ε = -L (dI/dt) = -2 x 10^-3 * e^(-t) (2t - t^2)

For the emf to be zero, we need:

ε = 0

This occurs when:

2t - t^2 = 0

Solving the Equation

Factoring the equation gives us:

t(2 - t) = 0

This results in two solutions:

• t = 0
• t = 2

Interpreting the Results

The first solution, t = 0, indicates the initial moment when the current starts flowing. The second solution, t = 2 seconds, is the time at which the emf becomes zero due to the current's behavior over time. Thus, the emf across the coil becomes zero at:

t = 2 seconds

In summary, the time at which the induced emf in the coil becomes zero is at 2 seconds, as derived from the relationship between current, inductance, and the rate of change of current. This illustrates how inductors respond to changing currents and the importance of understanding these dynamics in electrical circuits.