In an `LR`-circuit, time constant is that time in which current grows from zero to the value (where `I_(0)` is the steady state current)

Options

A. `0.63I_(0)`
B. `0.50I_(0)`
C. `0.37I_(0)`
D. `I_(0)`

Explanation

To solve the problem regarding the time constant in an LR circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the LR Circuit**:
An LR circuit consists of an inductor (L) and a resistor (R) connected in series. When a voltage is applied, the current (I) in the circuit does not immediately reach its maximum value (steady state current, I0) but increases gradually.

2. **Current Growth Equation**:
The current at any time \( t \) in an LR circuit is given by the formula:
\[ I(t) = I_0 \left(1 - e^{-\frac{R}{L}t}\right) \]
where:
- \( I(t) \) is the current at time \( t \),
- \( I_0 \) is the steady-state current,
- \( R \) is the resistance,
- \( L \) is the inductance,
- \( e \) is the base of the natural logarithm.

3. **Defining the Time Constant**:
The time constant \( \tau \) for an LR circuit is defined as:
\[ \tau = \frac{L}{R} \]
This is the time required for the current to reach approximately 63.2% of its maximum value (steady state).

4. **Finding Current at Time Constant**:
To find the current at the time constant \( \tau \), we substitute \( t = \tau \) into the current equation:
\[ I(\tau) = I_0 \left(1 - e^{-\frac{R}{L} \cdot \frac{L}{R}}\right) \]
Simplifying this gives:
\[ I(\tau) = I_0 \left(1 - e^{-1}\right) \]

5. **Calculating \( e^{-1} \)**:
The value of \( e^{-1} \) is approximately \( 0.3679 \). Therefore,
\[ I(\tau) = I_0 \left(1 - 0.3679\right) \approx I_0 \cdot 0.6321 \]
This means that at the time constant \( \tau \), the current reaches about 63.2% of its steady-state value \( I_0 \).

6. **Conclusion**:
The time constant \( \tau \) is the time it takes for the current to grow from zero to approximately \( 0.6321 I_0 \) in an LR circuit.

### Final Answer:
The time constant \( \tau \) in an LR circuit is defined as \( \tau = \frac{L}{R} \), and at this time, the current reaches approximately \( 0.6321 I_0 \).

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