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

Answer

To solve the problem of finding 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 (E) stored in an inductor is given by the formula:
\[ E = \frac{1}{2} L I^2 \]
where:
- \( E \) is the energy in joules,
- \( L \) is the inductance in henries (H),
- \( I \) is the current in amperes (A).

### Step 2: Differentiate the Energy Formula with Respect to Time
To find the energy stored in the inductor per unit time, we need to differentiate the energy formula with respect to time (t):
\[ \frac{dE}{dt} = \frac{d}{dt} \left( \frac{1}{2} L I^2 \right) \]

### Step 3: Apply the Chain Rule
Using the chain rule for differentiation, we have:
\[ \frac{dE}{dt} = \frac{1}{2} L \cdot 2I \cdot \frac{dI}{dt} \]
This simplifies to:
\[ \frac{dE}{dt} = L I \frac{dI}{dt} \]

### Step 4: Substitute Known Values
From the problem, we know:
- \( L = 2 \, \text{H} \) (inductance),
- \( I = 2 \, \text{A} \) (current),
- \( \frac{dI}{dt} = 4 \, \text{A/s} \) (rate of change of current).

Now, substituting these values into the equation:
\[ \frac{dE}{dt} = 2 \cdot 2 \cdot 4 \]

### Step 5: Calculate the Result
Now, calculate the value:
\[ \frac{dE}{dt} = 2 \cdot 2 \cdot 4 = 16 \, \text{J/s} \]

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

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