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

To determine the resultant inductance of a combination of inductors connected in parallel, we use the formula for inductors in parallel:

\( \frac{1}{L_{\text{total}}} = \frac{1}{L_1} + \frac{1}{L_2} + \ldots + \frac{1}{L_n} \)

For this problem, the coil of inductance \( L \) is divided into 6 equal parts. Therefore, each part has an inductance of \( \frac{L}{6} \).

When all 6 parts are connected in parallel, the formula becomes:

\( \frac{1}{L_{\text{total}}} = \frac{1}{\frac{L}{6}} + \frac{1}{\frac{L}{6}} + \frac{1}{\frac{L}{6}} + \frac{1}{\frac{L}{6}} + \frac{1}{\frac{L}{6}} + \frac{1}{\frac{L}{6}} \)

This is equivalent to:

\( \frac{1}{L_{\text{total}}} = 6 \times \frac{1}{\frac{L}{6}} = 6 \times \frac{6}{L} = \frac{36}{L} \)

Thus, the reciprocal of the total inductance is \(\frac{36}{L}\), so the total inductance is:

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

This shows that the resultant inductance of this combination is

\( \frac{L}{36} \)

Answer 2

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 \( \mathbf{\frac{L}{36}} \).