Question

a wire has resistance 12 ohms. It is bent in the form of a circle. The effective resistance between the two points on any diameter of the circle is,
a) 12 ohms
b) 24 ohms
c) 6 ohms
d) 3 ohms

Answer 1

By taking any diameter, the circumference would become half. As a result each half would have a resistance of 6Ω that are connected in parallel. So effective resistance would be 3Ω.

Answer 2

The effective resistance between two points on the circle is d) 3 ohm.  

Given data:

Resistance of the wire is 12 ohm. This wire is bent into circle. The total effective resistance on whole circled wire be R.

Then the circle can be two half circles with resistance each is $\frac{R}{2}$.  

Now the wire at any two points be connected via parallel to get the effective resistance on two points of the circle.

$Thereby\ \boldsymbol{R}_{E f f}=\frac{1}{\frac{R}{2}}+\frac{1}{\frac{R}{2}}=\frac{\left(\frac{R}{2}\right)\left(\frac{R}{2}\right)}{\left(\frac{R}{2}\right)+\left(\frac{R}{2}\right)}=\frac{\frac{R^{2}}{4}}{\frac{2 R}{2}}=\frac{R}{4}$

$\begin{array}{l}{R_{e f f}=\frac{R}{4}} \\ {R_{e f f}=\frac{12}{4}=3 \text { ohm. }}\end{array}$