A uniformly wound solenoid coil of self inductance `1.8xx10^(-4)H` and resistance `6Omega` is broken up into two identical coils. These identical coils are then connected in parallel across a `12V` battery of negligible resistance. The time constant and steady state current will be

Options

A. `0.1xx10^(-4)s`
B. `0.2xx10^(-4)s`
C. `0.3xx10^(-4)s`
D. `0.4xx10^(-4)s`

Answer

To solve the problem step by step, we will find the time constant and the steady state current for the given solenoid coil.

### Step 1: Find the new inductance of each coil
The original solenoid has a self-inductance \( L = 1.8 \times 10^{-4} \, H \). When it is broken into two identical coils, the inductance of each coil \( L' \) can be calculated as follows:

\[ L' = \frac{L}{2} = \frac{1.8 \times 10^{-4}}{2} = 0.9 \times 10^{-4} \, H \]

### Step 2: Find the equivalent inductance of the two coils in parallel
When two inductors are connected in parallel, the equivalent inductance \( L_{eq} \) is given by:

\[ \frac{1}{L_{eq}} = \frac{1}{L'} + \frac{1}{L'} = \frac{2}{L'} \]

Thus,

\[ L_{eq} = \frac{L'}{2} = \frac{0.9 \times 10^{-4}}{2} = 0.45 \times 10^{-4} \, H = 4.5 \times 10^{-5} \, H \]

### Step 3: Find the new resistance of each coil
The original resistance \( R = 6 \, \Omega \). When the coil is divided into two identical coils, the resistance of each coil \( R' \) is:

\[ R' = \frac{R}{2} = \frac{6}{2} = 3 \, \Omega \]

### Step 4: Find the equivalent resistance of the two coils in parallel
The equivalent resistance \( R_{eq} \) for two resistors in parallel is given by:

\[ \frac{1}{R_{eq}} = \frac{1}{R'} + \frac{1}{R'} = \frac{2}{R'} \]

Thus,

\[ R_{eq} = \frac{R'}{2} = \frac{3}{2} = 1.5 \, \Omega \]

### Step 5: Calculate the time constant
The time constant \( \tau \) for an RL circuit is given by:

\[ \tau = \frac{L_{eq}}{R_{eq}} \]

Substituting the values we found:

\[ \tau = \frac{4.5 \times 10^{-5}}{1.5} = 3.0 \times 10^{-5} \, s = 0.3 \times 10^{-4} \, s \]

### Step 6: Calculate the steady state current
The steady state current \( I \) can be calculated using Ohm's law:

\[ I = \frac{V}{R_{eq}} \]

Given \( V = 12 \, V \):

\[ I = \frac{12}{1.5} = 8 \, A \]

### Final Answers
- Time constant \( \tau = 0.3 \times 10^{-4} \, s \)
- Steady state current \( I = 8 \, A \)

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