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 ms
B. 20 ms
C. 35 ms
D. 1 ms

Explanation

To solve the problem step-by-step, we will use the concepts of inductance, resistance, and the behavior of current in an LR circuit 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) = 1.0 A (approximately at a certain time)

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

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

### Step 4: Write the equation for current in the circuit
The current (I(t)) at any time t in an LR circuit is given by:
\[ I(t) = I₀ \left(1 - e^{-\frac{t}{τ}}\right) \]
Substituting I₀ and τ:
\[ I(t) = 2 \left(1 - e^{-\frac{t}{1.4 \times 10^{-3}}}\right) \]

### Step 5: Set I(t) to the given current (1.0 A)
We set the equation equal to the given current:
\[ 1 = 2 \left(1 - e^{-\frac{t}{1.4 \times 10^{-3}}}\right) \]

### Step 6: Solve for e-term
Rearranging the equation:
\[ \frac{1}{2} = 1 - e^{-\frac{t}{1.4 \times 10^{-3}}} \]
\[ e^{-\frac{t}{1.4 \times 10^{-3}}} = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 7: Take the natural logarithm
Taking the natural logarithm of both sides:
\[ -\frac{t}{1.4 \times 10^{-3}} = \ln\left(\frac{1}{2}\right) \]
\[ -\frac{t}{1.4 \times 10^{-3}} = -0.693 \]

### Step 8: Solve for t
Multiplying both sides by -1.4 × 10^(-3):
\[ t = 1.4 \times 10^{-3} \times 0.693 \]
\[ t \approx 0.97 \times 10^{-3} \, \text{s} = 0.97 \, \text{ms} \]

### Step 9: Round the answer
Rounding to two significant figures gives:
\[ t \approx 1 \, \text{ms} \]

### Final Answer
The time at which the current in the coil is approximately 1.0 A is **1 ms**.

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