During current growth in an L-R circuit the time constant is the time in which the magnitude of current becomes

Options

A. `i_(0)`
B. `i_(0)//2`
C. `0.63i_(0)`
D. `0.37i_(0)`

Explanation

To solve the question regarding the time constant in an L-R circuit, we will go through the following steps:

### Step 1: Understand the L-R Circuit
An L-R circuit consists of an inductor (L) and a resistor (R) connected in series with a battery of electromotive force (E). The inductor stores energy in the magnetic field when current flows through it, while the resistor dissipates energy as heat.

**Hint:** Remember that the inductor's behavior is characterized by its inductance (L) and the resistor's behavior is characterized by its resistance (R).

### Step 2: Define the Current in the Circuit
The current (I) at any time (t) in the L-R circuit can be expressed by the formula:
\[ I(t) = I_0 \left(1 - e^{-\frac{R t}{L}}\right) \]
where:
- \( I_0 \) is the maximum current (steady-state current) given by \( I_0 = \frac{E}{R} \),
- \( e \) is the base of the natural logarithm.

**Hint:** The term \( e^{-\frac{R t}{L}} \) represents the exponential decay of the current growth in the circuit.

### Step 3: Define the Time Constant
The time constant (\( \tau_L \)) for an L-R circuit is defined as:
\[ \tau_L = \frac{L}{R} \]
This time constant indicates the time it takes for the current to reach approximately 63.2% of its maximum value (\( I_0 \)).

**Hint:** The time constant is a crucial parameter that helps in understanding how quickly the circuit responds to changes in voltage.

### Step 4: Calculate the Current at One Time Constant
To find the current at one time constant (\( t = \tau_L \)):
\[ I(\tau_L) = I_0 \left(1 - e^{-1}\right) \]
Since \( e^{-1} \) is approximately 0.3679, we can calculate:
\[ I(\tau_L) = I_0 \left(1 - 0.3679\right) \]
\[ I(\tau_L) \approx I_0 \times 0.6321 \]

**Hint:** The value \( 1 - e^{-1} \) gives you the fraction of the maximum current reached after one time constant.

### Step 5: Conclusion
After one time constant, the magnitude of the current becomes approximately 0.63 times the initial current \( I_0 \). Thus, the correct answer to the question is:
\[ \text{Current at } \tau_L = 0.63 I_0 \]

**Final Answer:** The time constant is the time in which the magnitude of current becomes \( 0.63 I_0 \).