Final answer:
The question asks for the voltage across a 10-mH inductor at t=6 seconds given a current formula. The voltage is found by differentiating the current with respect to time and multiplying by the inductance. Power is then calculated using the voltage found and the current at t=6 seconds.
Step-by-step explanation:
The question involves calculating the voltage across an inductor and the power at a specific time given the time-varying current. According to Faraday's Law of Induction, the voltage (V) induced across an inductor can be calculated using the formula V = L (di/dt), where L is the inductance and di/dt is the rate of change of current with respect to time. In this case, the current I(t) = 10e⁻₉/² is provided, and we need to find the voltage at time t = 6 s.
First, we need to calculate di/dt at t = 6 s. By differentiating I(t) with respect to time, we get di/dt = -5e⁻₉/². At t = 6 s, this derivative becomes di/dt = -5e⁻₁₂. Then, using the given inductance L = 10 mH = 0.01 H, we calculate the voltage V = L (di/dt). So, V = 0.01 H * (-5e⁻₁₂ A/s), and after evaluating the exponent and multiplication, we can find the voltage in millivolts (mV).
To find the power P at t = 6 s, we use the formula P = V * I, where I is the current at t = 6 s. With the voltage already found, we calculate the current I(6) using the given I(t) formula, and then calculate the power.