A coil of inductance 8.4 mH and resistance `6 (Omega)` is connected to a 12 V battery. The current in the coil is 1.0 A at approximately the time

Options

A. `500 s`
B. `25 s`
C. `35 ms`
D. `1 ms`

Explanation

To solve the problem step-by-step, we will use the formula for the current in an inductor connected to a DC voltage source.

### Step 1: Identify the given values
- Inductance (L) = 8.4 mH = 8.4 × 10^(-3) H
- Resistance (R) = 6 Ω
- Voltage (V) = 12 V
- Current (I) = 1 A

### Step 2: Calculate the maximum current (I₀)
The maximum current in the circuit can be calculated using Ohm's law:
\[ I_0 = \frac{V}{R} \]
Substituting the values:
\[ I_0 = \frac{12 \, \text{V}}{6 \, \Omega} = 2 \, \text{A} \]

### Step 3: Calculate the time constant (τ)
The time constant (τ) for an RL circuit is given by:
\[ \tau = \frac{L}{R} \]
Substituting the values:
\[ \tau = \frac{8.4 \times 10^{-3} \, \text{H}}{6 \, \Omega} = 1.4 \times 10^{-3} \, \text{s} \]

### Step 4: Use the current equation for an RL circuit
The equation for the current in the inductor at time t is given by:
\[ I(t) = I_0 \left(1 - e^{-\frac{t}{\tau}}\right) \]
We know that at time t, the current I(t) is 1 A. Therefore:
\[ 1 = 2 \left(1 - e^{-\frac{t}{\tau}}\right) \]

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ \frac{1}{2} = 1 - e^{-\frac{t}{\tau}} \]
This simplifies to:
\[ e^{-\frac{t}{\tau}} = \frac{1}{2} \]

### Step 6: Take the natural logarithm
Taking the natural logarithm of both sides:
\[ -\frac{t}{\tau} = \ln\left(\frac{1}{2}\right) \]
This can be rewritten as:
\[ t = -\tau \ln\left(\frac{1}{2}\right) \]

### Step 7: Substitute τ into the equation
Substituting the value of τ:
\[ t = -1.4 \times 10^{-3} \, \text{s} \cdot \ln\left(\frac{1}{2}\right) \]
Using the value of \(\ln\left(\frac{1}{2}\right) \approx -0.6931\):
\[ t = 1.4 \times 10^{-3} \cdot 0.6931 \]

### Step 8: Calculate the time
Calculating the value:
\[ t \approx 0.9703 \times 10^{-3} \, \text{s} \]
This can be expressed as:
\[ t \approx 1 \, \text{ms} \]

### Conclusion
The time at which the current in the coil reaches approximately 1 A is about 1 millisecond.