Animated Solution for Physics - Current Electricity: If in the circuit, power dissipation is $150\text{ W}$, then $R$ is

Imagine you are an electrical detective, and you've just been handed a blueprint of a circuit. Your mission? To unmask the identity of a hidden resistor, $R$.

Let's look at the clues provided. We have a battery supplying a steady $15\text{ V}$ to the circuit. Connected to this battery are two resistors arranged in parallel: one is a known $2\Omega$ resistor, and the other is our mysterious $R$. The final, most crucial piece of evidence is that the entire circuit dissipates exactly $150\text{ W}$ of power.

Power and Resistance

Whenever you are given the total power and the total voltage of a circuit, you have a direct key to unlock the total equivalent resistance. The relationship that binds them together is the power formula:

Here, $P$ is the total power dissipated, $V$ is the voltage across the combination, and $R_{\text{eq}}$ is the equivalent resistance of the entire parallel network.

Let's substitute our known clues into this master equation. We plug in $150\text{ W}$ for power and $15\text{ V}$ for voltage:

Now, it's just a matter of simple algebra. We know that $15^2$ is $225$. Rearranging the equation to solve for $R_{\text{eq}}$, we get:

Dividing both numbers, we find that the equivalent resistance of the entire circuit is exactly $1.5\Omega$ (or $\frac{3}{2}\Omega$).

Now that we know the total resistance is $1.5\Omega$, we can zoom in on the parallel branches. The rule for finding the equivalent resistance of resistors in parallel is:

We know $R_{\text{eq}}=1.5\Omega$, $R_1=2\Omega$, and $R_2=R$. Let's substitute these values into our parallel formula. Since $1.5$ is $\frac{3}{2}$, its reciprocal is $\frac{2}{3}$:

To isolate our unknown $R$, we simply subtract $\frac{1}{2}$ from both sides:

To subtract these fractions, we find a common denominator, which is $6$:

Taking the reciprocal of both sides reveals the true identity of our hidden resistor:

And there you have it! By systematically applying the power formula and the rules of parallel circuits, we've successfully solved the mystery. The correct option is (b).