Question

A coil of inductance \( L \) is divided into 6 equal parts. All these parts are connected in parallel. The resultant inductance of this combination is:

• A. \( \frac{L}{6} \)
• B. \( \frac{L}{36} \)
• C. \( \frac{L}{24} \)
• D. \( 6L \)

Hint:

For inductors in parallel, use the formula: \[ \frac{1}{L_{\text{eq}}} = \sum \frac{1}{L_i}. \] If \( n \) identical inductors \( L/n \) are connected in parallel, the total inductance is further divided by \( n \).

Answer 1

The Correct Option is B

Inductance in Parallel Problem

Step 1: Dividing the inductance into 6 equal parts

The total inductance \( L \) is divided into 6 equal parts. Inductance of each part is calculated as follows:

\( L_{\text{each}} = \frac{L}{6} \)

Step 2: Determining the effective inductance for parallel combination

In a parallel combination of inductances, the reciprocal of the effective inductance is given by:

\( \frac{1}{L_{\text{eq}}} = \frac{1}{L_1} + \frac{1}{L_2} + \cdots + \frac{1}{L_6} \)

Since all inductances are equal to \( L_{\text{each}} = \frac{L}{6} \), this becomes:

\( \frac{1}{L_{\text{eq}}} = 6 \times \frac{1}{\frac{L}{6}} \)

Step 3: Simplifying the equation

\( \frac{1}{L_{\text{eq}}} = 6 \times \frac{6}{L} \)

\( \frac{1}{L_{\text{eq}}} = \frac{36}{L} \)

Now take the reciprocal to obtain the effective inductance:

\( L_{\text{eq}} = \frac{L}{36} \)

Step 4: Final Answer

Thus, the resultant inductance of the combination is \( \mathbf{\frac{L}{36}} \).