An inductance coil have the time constant 4 sec, if it is cut into two equal parts and connected parallel then new time constant of the circuit :-

Options

A. 4 sec
B. 2 sec
C. 1 sec
D. 0.5 sec

Answer

To solve the problem step by step, we need to analyze the situation involving the inductance coil, its time constant, and the effects of cutting it into two equal parts and connecting them in parallel.

### Step-by-Step Solution:

1. **Understand the Time Constant**:
The time constant (τ) for an inductor is given by the formula:
\[ \tau = \frac{L}{R} \]
where \(L\) is the inductance and \(R\) is the resistance.

2. **Given Information**:
We know that the time constant of the original coil is:
\[ \tau = 4 \text{ seconds} \]

3. **Cutting the Coil**:
When the coil is cut into two equal parts, the inductance \(L\) of each part becomes half of the original inductance:
\[ L' = \frac{L}{2} \]

4. **Resistance of Each Part**:
Assuming the resistance is uniformly distributed, when the coil is cut into two equal parts, the resistance of each part also becomes half:
\[ R' = \frac{R}{2} \]

5. **Connecting in Parallel**:
When the two halves are connected in parallel, the equivalent resistance \(R_{eq}\) of two resistors in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R'} + \frac{1}{R'} = \frac{2}{R'} \]
Thus,
\[ R_{eq} = \frac{R'}{2} = \frac{R/2}{2} = \frac{R}{4} \]

6. **New Time Constant**:
Now, we can find the new time constant \(τ_{new}\) using the new inductance and equivalent resistance:
\[ \tau_{new} = \frac{L'}{R_{eq}} = \frac{L/2}{R/4} = \frac{L}{2} \cdot \frac{4}{R} = \frac{4L}{2R} = \frac{4L}{R} \cdot \frac{1}{2} = 4 \text{ seconds} \]

7. **Conclusion**:
The new time constant after cutting the coil into two equal parts and connecting them in parallel remains the same:
\[ \tau_{new} = 4 \text{ seconds} \]

### Final Answer:
The new time constant of the circuit is **4 seconds**.

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