Question

A uniform wire of resistance \(R\) is cut into four equal parts. These parts are then connected in parallel. The equivalent resistance of the combination is:

• A. \(R\)
• B. \(\dfrac{R}{4}\)
• C. \(\dfrac{R}{16}\)
• D. \(4R\)

Hint:

If a wire of resistance \(R\) is cut into \(n\) equal parts: \[ R_{\text{each}}=\frac{R}{n} \] If all \(n\) pieces are connected in parallel: \[ R_{\text{eq}} = \frac{R}{n^2} \] For \(n=4\): \[ R_{\text{eq}} = \frac{R}{16} \]

Answer 1

The Correct Option is C

Step 1: Find resistance of each part. Resistance is proportional to length. When the wire is cut into four equal parts: \[ R_{\text{each}} = \frac{R}{4} \]

Step 2: Connect the four parts in parallel. For \(n\) identical resistors \(r\) connected in parallel: \[ R_{\text{eq}} = \frac{r}{n} \] Here, \[ r=\frac{R}{4} \] and \[ n=4 \] Therefore, \[ R_{\text{eq}} = \frac{\frac{R}{4}}{4} \] \[ = \frac{R}{16} \]

Step 3: Identify the correct option. \[ \boxed{R_{\text{eq}}=\frac{R}{16}} \]