Animated Solution for Physics - Current Electricity: A heater is designed to operate with a power of $1000\text{ W}$ in a $100\text{ V}$ line. It is connected in combination with a resistance of $10\Omega$ and a resistance $R$, to a $100\text{ V}$ mains as shown in the figure. What will be the value of $R$ so that the heater operates with a power of $62.5\text{ W}$ ?

Let's dive into this fascinating circuit problem. We are presented with a heater that is designed to operate at a power of $1000\text{ W}$ when connected to a $100\text{ V}$ line. However, in our specific circuit, it's not connected directly to the $100\text{ V}$ source. Instead, it's placed in parallel with an unknown resistor $R$, and this entire parallel combination is in series with a $10\Omega$ resistor.

The goal is to find the exact value of $R$ that forces the heater to operate at a reduced power of $62.5\text{ W}$. To solve this, we need to systematically break down the circuit using Ohm's Law and power formulas.

The very first step is to determine the intrinsic resistance of the heater. The resistance of a device is a physical property that remains constant (assuming temperature effects are negligible), regardless of the voltage applied to it. We can find this using its rated specifications:

Substituting the given rated values, we get:

Now that we know the heater acts as a $10\Omega$ resistor, we can find out how much current is actually flowing through it when it operates at $62.5\text{ W}$. We use the power formula relating current and resistance:

So, exactly $2.5\text{ A}$ of current must flow through the heater branch.

Next, we need to express the main current $I$ drawn from the $100\text{ V}$ source. The equivalent resistance of the circuit is the series combination of the $10\Omega$ resistor and the parallel block:

Using Ohm's law, the main current $I$ is:

When this main current reaches the parallel junction, it splits. According to the current divider rule, the current through the heater ($i_H$) is:

Substituting our known values and expressions:

Notice how beautifully the $(10+R)$ terms cancel out from the numerator and denominator! We are left with a much simpler equation:

Cross-multiplying yields:

And there we have it! The unknown resistance $R$ must be exactly $5\Omega$ to achieve the desired power output in the heater.