Animated Solution for Physics - Current Electricity: A heater coil is cut into two equal parts and only one part is now used in the heater. The heat generated will now be

Imagine it is a freezing winter night, and you are sitting next to a glowing room heater. Inside that heater is a metallic coil, resisting the flow of electricity and radiating warmth. Now, let us pose a fascinating thought experiment: What would happen if you took that coil, cut it exactly in half, and plugged just one of those halves back into the wall?

Your intuition might scream that a smaller coil means less heat. After all, there is less material to get hot, right? Surprisingly, the laws of physics dictate the exact opposite. Let us dive into the mathematics of Joule heating to uncover why.

When you plug an appliance into a wall socket, the power grid provides a constant voltage (let us call it $V$). The current that flows through the appliance is entirely dependent on the appliance's resistance.

To calculate the heat $H$ generated over a time $t$, we use Joule's Law of Heating. While $H=I^{2}Rt$ is a famous form of this law, it is a trap in this scenario because the current $I$ will change when we alter the coil. Instead, we must use the form that relies on our constant voltage:

This is our master equation. It tells us exactly how much heat the original, unbroken coil produces.

Now, let us analyze the act of cutting the coil. The resistance $R$ of a uniform wire is directly proportional to its length $L$ ($R\propto L$).

If we cut the coil into two equal halves, the length of the piece we are using is now $L/2$. Consequently, its new resistance $R^{\prime }$ will be exactly half of the original resistance:

We are now ready to calculate the new heat generated, $H_2$, by this shorter coil. We substitute our new resistance $R^{\prime }$ into our master equation:

Substituting $R^{\prime }=R/2$, we get:

By simple algebra, the $2$ in the denominator of the fraction flips up to the numerator:

Look closely at the term inside the parentheses. It is exactly our original heat, $H_1$! Therefore:

The heat generated is doubled! By halving the resistance, the coil draws twice as much current from the constant voltage source. Since power is the product of voltage and current ($P=VI$), doubling the current doubles the power output.

While it might keep you twice as warm, be warned: drawing twice the current might exceed the wire's physical limits, causing it to melt and snap. Physics is elegant, but engineering requires caution!