The current through a 10-mH inductor is 10e−t∕2 A. Find the voltage and the power at t = 8 s.

The voltage across an inductor can be determined by the formula:

V_L(t) = L * (di/dt),

where V_L(t) is the voltage across the inductor at time t, L is the inductance, di/dt is the derivative of the current with respect to time.

Given: L = 10 mH (or 0.01 H) i(t) = 10e^(-t/2) A

Let's first find the derivative of the current with respect to time (t):

di/dt = d/dt [10e^(-t/2)] = -1/2 * 10e^(-t/2) = -5e^(-t/2)

Now we can use this derivative to find the voltage across the inductor at t = 8 s:

V_L(8) = L * (di/dt at t = 8s) = 0.01 H * (-5e^(-8/2)) = 0.01 * (-5e^(-4)) V

Using the approximate value of e^(-1) = 0.3679:

V_L(8) ≈ 0.01 * (-5 * 0.3679^4) V_L(8) ≈ 0.01 * (-5 * 0.0183) (since 0.3679^4 is approximately equal to 0.0183) V_L(8) ≈ -0.00915 V

Therefore, the voltage across the inductor at t = 8 seconds is approximately -0.00915 volts.

Next, we calculate the power at time t = 8 s using the power formula P(t) = V(t) * I(t):

P(8) = V_L(8) * i(8) = (-0.00915 V) * (10e^(-8/2) A)

Using the same approximation for e^(-4):

P(8) ≈ (-0.00915) * (10 * 0.0183) P(8) ≈ (-0.00915) * (0.183) P(8) ≈ -0.00167445 W

Therefore, the power at t = 8 seconds is approximately -0.00167445 watts, indicating that the inductor is actually absorbing power from the circuit.

Extra: Inductors are passive components in an electrical circuit that store energy in the form of a magnetic field. When the current through an inductor changes, the magnetic field also changes and consequently induces a voltage across the inductor. The relationship between the induced voltage and the rate of change of current is given by Faraday's Law of electromagnetic induction, which is the basis for the formula V_L(t) = L * (di/dt).

In the calculation above, we found that the voltage is negative at t = 8 seconds. This is because the current is decreasing at this time (the exponential is fading with time), which means the inductor is resisting the decrease by inducing a voltage that tries to keep the current going in the same direction (Lenz's Law).

The power calculated as negative also implies that the inductor is not dissipating power as a resistor would, but rather storing energy in its magnetic field. When power is negative, it means the component is absorbing power rather than supplying it, which is characteristic of inductive and capacitive elements during portions of the AC cycle or during transient periods in DC circuits.