A current of `2A` is increasing at a rate of `4A//s` through a coil of inductance `2H`. The energy stored in the inductor per unit time is

Options

A. 2 J/s
B. 1 J/s
C. 16 J/s
D. 4 J/s

Answer

To find the energy stored in the inductor per unit time, we can follow these steps:

### Step 1: Understand the formula for energy stored in an inductor
The energy (U) stored in an inductor is given by the formula:
\[ U = \frac{1}{2} L I^2 \]
where:
- \( U \) is the energy in joules,
- \( L \) is the inductance in henries (H),
- \( I \) is the current in amperes (A).

### Step 2: Differentiate the energy with respect to time
To find the rate of change of energy with respect to time (which is the energy stored per unit time), we differentiate the energy formula:
\[ \frac{dU}{dt} = \frac{d}{dt} \left( \frac{1}{2} L I^2 \right) \]
Using the chain rule, we have:
\[ \frac{dU}{dt} = \frac{1}{2} L \cdot 2I \cdot \frac{dI}{dt} = L I \frac{dI}{dt} \]

### Step 3: Substitute the given values
From the problem, we have:
- \( L = 2 \, \text{H} \)
- \( I = 2 \, \text{A} \)
- \( \frac{dI}{dt} = 4 \, \text{A/s} \)

Now substituting these values into the differentiated equation:
\[ \frac{dU}{dt} = L I \frac{dI}{dt} = 2 \cdot 2 \cdot 4 \]

### Step 4: Calculate the result
Now we calculate:
\[ \frac{dU}{dt} = 2 \cdot 2 \cdot 4 = 16 \, \text{J/s} \]

### Final Answer
The energy stored in the inductor per unit time is:
\[ \frac{dU}{dt} = 16 \, \text{J/s} \]