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 resulting curve?

Options

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Answer

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

### Step 2: Determine the Rate of Energy Storage
The rate at which energy is stored in the inductor can be expressed as:
\[ \text{Rate} = \frac{dU}{dt} = \frac{d}{dt} \left( \frac{1}{2} L I^2 \right) \]
Using the chain rule, we can express this as:
\[ \frac{dU}{dt} = L I \frac{dI}{dt} \]
This indicates that the rate of energy storage depends on both the current \( I \) and the rate of change of current \( \frac{dI}{dt} \).

### Step 3: Analyze Current Growth Over Time
In an LR circuit connected to a battery, the current \( I \) grows according to the equation:
\[ I(t) = I_0 (1 - e^{-\frac{R}{L}t}) \]
where \( I_0 \) is the maximum current, \( R \) is the resistance, and \( t \) is time.

### Step 4: Determine Behavior at Key Points
- At \( t = 0 \):
- \( I = 0 \) → Rate of energy storage \( = 0 \)
- As \( t \) increases:
- \( I \) increases → Rate of energy storage increases.
- At \( t \to \infty \):
- \( I \) approaches \( I_0 \) → Rate of energy storage \( = 0 \) again (since \( \frac{dI}{dt} \to 0 \)).

### Step 5: Sketch the Curve
From the analysis:
- The curve starts at \( (0, 0) \) when \( t = 0 \).
- It rises to a maximum value as \( I \) increases, and then it falls back to \( 0 \) as \( t \to \infty \).
- The resulting curve is a concave shape that starts at zero, rises to a peak, and then returns to zero.

### Conclusion
The best representation of the curve plotting the rate of energy stored in the inductor against time is a curve that starts at zero, rises to a maximum, and then returns to zero.