In an `LR` circuit connected to a battery, the rate at which energy is stored in the inductor is plotted against time during the growth of current in the circuit. Which of the following best represents the resulation curve?

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the behavior of energy stored in an inductor in an LR circuit over time as the current grows. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Energy Stored in the Inductor
The energy (U) stored in an inductor is given by the formula:
\[ U = \frac{1}{2} L I^2 \]
where \( L \) is the inductance and \( I \) is the current flowing through the inductor.

### Step 2: Determine the Rate of Change of Energy
To find the rate at which energy is stored in the inductor, we need to differentiate the energy with respect to time (t):
\[ \frac{dU}{dt} = \frac{d}{dt} \left( \frac{1}{2} L I^2 \right) \]
Using the chain rule, we get:
\[ \frac{dU}{dt} = L I \frac{dI}{dt} \]
This expression indicates that the rate of change of energy depends on both the current (I) and the rate of change of current (\( \frac{dI}{dt} \)).

### Step 3: Analyze the Behavior at Initial and Final Times
- At \( t = 0 \): The current \( I \) is zero, hence:
\[ \frac{dU}{dt} = L \cdot 0 \cdot \frac{dI}{dt} = 0 \]
- At \( t = \infty \): The current reaches its maximum value \( I_0 \), and the rate of change of current \( \frac{dI}{dt} \) approaches zero as the circuit reaches steady state. Thus:
\[ \frac{dU}{dt} = L I_0 \cdot 0 = 0 \]

### Step 4: Sketch the Graph
From the analysis, we can conclude:
- The rate of energy storage starts at zero when \( t = 0 \).
- It increases as the current increases and reaches a maximum value at some point during the growth of current.
- Finally, as the current stabilizes, the rate of energy storage returns to zero when \( t \) approaches infinity.

### Step 5: Identify the Correct Graph
Based on the above behavior, the graph that best represents the rate of energy storage against time will start at zero, rise to a maximum, and then return to zero as time approaches infinity.

The correct option is **Graph A**, which shows this behavior.