Question

The voltage applied to a 212 mH inductor is given by $V(t)=15e^{-5t}$ V. Calculate the current

• A. \( 16.782e^{-10t} \)
• B. \( 15.75e^{-5t} \)
• C. \( 11.27e^{-10t} \)
• D. \( 14.15e^{-5t} \)

Hint:

Remember the fundamental inductor equation: $V(t) = L \frac{dI(t)}{dt}$. To find current from voltage, you need to integrate $I(t) = \frac{1}{L} \int V(t) dt$. Pay close attention to units (mH needs to be converted to H) and the constant of integration. For exponential functions, $\int e^{ax} dx = \frac{1}{a} e^{ax}$. In multiple-choice questions involving magnitude, ensure your calculated numerical value matches one of the options.

Answer 1

The Correct Option is D

To solve for the current through an inductor given the voltage across it, we use the inductor's voltage-current relationship.

1. Understanding the Concepts:

- Inductor Voltage-Current Relationship: The voltage across an inductor is given by \( V(t) = L \frac{di(t)}{dt} \), where \( L \) is the inductance and \( i(t) \) is the current.
- Inductance (L): The property of an inductor to oppose changes in current, measured in Henrys (H).
- Voltage (V(t)): The potential difference across the inductor as a function of time.
- Current (i(t)): The flow of charge through the inductor as a function of time.

2. Given Values:

\( V(t) = 15e^{-5t} \text{ V} \)
\( L = 212 \text{ mH} = 0.212 \text{ H} \)

3. Calculating the Current:

We need to find \( i(t) \) given \( V(t) = L \frac{di(t)}{dt} \). Rearranging for \( di(t) \) gives us \( di(t) = \frac{V(t)}{L} dt \). Now, we integrate both sides with respect to time:

\( i(t) = \int \frac{V(t)}{L} dt = \int \frac{15e^{-5t}}{0.212} dt = \frac{15}{0.212} \int e^{-5t} dt \)

\( i(t) = \frac{15}{0.212} \cdot \frac{e^{-5t}}{-5} + C = -\frac{15}{0.212 \times 5} e^{-5t} + C = -\frac{3}{0.212} e^{-5t} + C \)

\( i(t) = -14.15 e^{-5t} + C \)

Assuming the initial current \( i(0) = 0 \), then \( 0 = -14.15 e^{0} + C \)
so \( C = 14.15 \). Therefore, \( i(t) = -14.15e^{-5t} + 14.15 = 14.15(1-e^{-5t}) \) 

Final Answer:

Based on the options, is: \( 14.15e^{-5t} \) . If the intial condition is i(0) = 0, then the final answer would be $14.15(1-e^{-5t})$.