Animated Solution for Physics - Current Electricity: A wire when connected to $220\text{ V}$ mains supply has power dissipation $P_1$. Now, the wire is cut into two equal pieces which are connected in parallel to the same supply. Power dissipation in this case is $P_2$. Then, $P_2:P_1$ is

A Single Wire

Imagine a single, uniform wire connected across a constant voltage supply, like the $220\text{ V}$ mains in our homes. This wire has a certain resistance, let's call it $R$. When current flows through it, it dissipates electrical energy as heat.

The power dissipated by a resistor connected across a constant voltage $V$ is given by the formula $P=\frac{V^2}{R}$. We specifically choose this form of the power equation (instead of $P=I^{2}R$) because the voltage $V$ remains constant throughout our experiment. So, for our initial setup, the power dissipation is simply $P_1=\frac{V^2}{R}$.

Cutting and Reconnecting

Now, let's perform a little surgery on our circuit. We take the exact same wire and cut it perfectly in half. What happens to the resistance?

We know that the resistance of a wire is directly proportional to its length ($R\propto l$). By cutting the wire in half, we have halved its length, which means the resistance of each individual piece is now exactly half of the original resistance, or $\frac{R}{2}$.

Next, we take these two $\frac{R}{2}$ pieces and connect them in parallel across the same $220\text{ V}$ supply. In a parallel circuit, the equivalent resistance $R_{eq}$ drops significantly. Using the parallel combination formula, $\frac{1}{R_{eq}}=\frac{1}{R/2}+\frac{1}{R/2}$, we find that $R_{eq}=\frac{R}{4}$. The overall resistance of the circuit has plummeted to one-fourth of its original value!

Power Ratio

With our new, much lower equivalent resistance, let's calculate the new power dissipation, $P_2$.

Since the voltage is still the same $V$, we plug our new resistance into the power formula: $P_2=\frac{V^2}{R_{eq}}=\frac{V^2}{R/4}$. A little algebraic rearrangement brings the $4$ to the numerator, giving us $P_2=4\frac{V^2}{R}$.

Finally, we want to find the ratio of the new power to the old power, $P_2:P_1$.

The power dissipation has increased by a factor of 4! This is a beautiful illustration of how manipulating physical dimensions and circuit configurations can drastically alter the energy output of a system. If you cut a wire into $n$ equal pieces and connect them in parallel, the power dissipation will always increase by a factor of $n^2$.