Animated Solution for Physics - Current Electricity: Three resistances of equal value are arranged in the different combinations shown below. Arrange them in increasing order of power dissipation

Imagine you are an electrical engineer tasked with designing a heating element. You have three identical resistors, each with a resistance of $r$, and you can wire them up in various ways. The problem presents us with four distinct circuit configurations (I, II, III, and IV).

The crucial piece of information hidden in plain sight within the diagrams is the current $i$. Notice how the same total current $i$ is shown entering each of the four combinations. Our mission is to determine which configuration dissipates the least power and which dissipates the most, arranging them in increasing order.

To compare the power dissipated by these different circuits, we need the right mathematical tool. Recall the fundamental formulas for electrical power: $P=V\cdot i$, $P=\frac{V^2}{R}$, and $P=i^{2}R$.

Which one should we use? Since the problem explicitly shows that the total current $i$ is constant across all four circuits, the most direct and reliable formula is $P=i^2{R}_{\text{eq}}$. Because $i$ is the same for all, the power dissipated is directly proportional to the equivalent resistance of the circuit. Mathematically, we can write this as $P\propto R_{\text{eq}}$.

This simplifies our task immensely: the circuit with the highest equivalent resistance will dissipate the most power, and the one with the lowest resistance will dissipate the least.

Let's roll up our sleeves and calculate the equivalent resistance for each circuit one by one.

Circuit I: Here, all three resistors are connected end-to-end in a single path. This is a simple series combination. We just add them up:

Circuit II: The current splits into two branches. The top branch has just one resistor of resistance $r$. The bottom branch has two resistors in series, making its resistance $2r$. These two branches are in parallel. Using the product-over-sum rule, the equivalent resistance is:

Circuit III: In this configuration, the current splits into three separate branches, each containing a single resistor. This means all three resistors are in parallel. For three identical resistors in parallel, the equivalent resistance is simply the resistance of one divided by three:

Circuit IV: First, the current goes through a parallel combination of two resistors. The resistance of this part is $\frac{r}{2}$. Then, this combination is in series with the third resistor. So, we add $r$ to $\frac{r}{2}$:

Now we have all the pieces of the puzzle. Let's list the values and compare them carefully to avoid any silly mistakes with fractions: - $R_{III}=\frac{r}{3}\approx 0.33r$ - $R_{II}=\frac{2r}{3}\approx 0.67r$ - $R_{IV}=\frac{3r}{2}=1.50r$ - $R_I=3r=3.00r$

Arranging them in increasing order of resistance, we get:

Since we established earlier that power dissipation is directly proportional to the equivalent resistance ($P\propto R_{\text{eq}}$), the order of power dissipation will be exactly the same as the order of resistance:

This perfectly matches option (a).

A Quick Thought Experiment: What if, instead of a constant current, a constant voltage $V$ was applied across each of these circuits? In that case, we would use the formula $P=\frac{V^2}{R_{\text{eq}}}$. Power would be inversely proportional to resistance ($P\propto \frac{1}{R_{\text{eq}}}$). The circuit with the lowest resistance would dissipate the most power, completely reversing our final order! Always read the question carefully to see what is kept constant.