Animated Solution for Physics - Current Electricity: A square shaped wire with resistance of each side $3\Omega$ is bent to form a complete circle. The resistance between two diametrically opposite points of the circle in unit of $\Omega$ will be

The problem presents us with a classic scenario of reshaping a conductor and observing how its electrical properties adapt. We start with a square-shaped wire and transform it into a circle, ultimately finding the equivalent resistance between two specific points. Let's break down the physics step-by-step.

Imagine you are holding a square-shaped wire. The problem states that each of its four sides has a resistance of $3\Omega$.

Since the sides are connected end-to-end, they form a single continuous path. The total resistance of this entire wire is simply the sum of the resistances of all four sides.

This $12\Omega$ represents the intrinsic resistance of the entire length of the wire, regardless of how it is shaped, as long as we don't stretch or compress it.

Now, we take this exact same wire and bend it to form a complete circle.

Remember a fundamental principle: bending a wire does not change its total length, its cross-sectional area, or the material it is made of. According to the formula for resistance, $R=\rho \frac{l}{A}$, since none of these parameters change, the total resistance of the full circle remains exactly the same.

The question asks for the equivalent resistance between two diametrically opposite points. Let's mark these points as $A$ and $B$ on the circle.

Notice how these two points perfectly slice the circle into two identical halves: an upper semicircle and a lower semicircle. Since resistance is directly proportional to the length of the wire ($R\propto l$), each semicircle, being exactly half the total length, will have exactly half the total resistance.

If we were to connect a battery across points $A$ and $B$, the current would split into two distinct paths: the upper semicircle and the lower semicircle. This means our two $6\Omega$ resistors are effectively connected in parallel.

For two resistors connected in parallel, the equivalent resistance $R_{eq}$ is given by the product over sum formula:

Let's substitute our values into this master equation:

And there we have it! The equivalent resistance between the diametrically opposite points is $3\Omega$. A beautiful transformation from a square to a circle, elegantly solved using the principles of parallel circuits.