Animated Solution for Physics - Current Electricity: The equivalent resistance between points $A$ and $B$ of the circuit given below is ....... .

When you first look at this circuit, it seems designed to confuse you. Wires cross over each other, connecting different parts of the resistor chain in a way that doesn't immediately look like a standard series or parallel combination. However, in physics, appearances can be deceiving. The key to unraveling this tangled web lies in a fundamental concept: equipotential nodes.

An ideal connecting wire has zero resistance. According to Ohm's law ($V=IR$), if the resistance $R$ is zero, the potential difference $V$ across the wire must also be zero, regardless of the current flowing through it. This means that every single point along an uninterrupted, resistanceless wire is at the exact same electrical potential.

Let's apply this concept to our circuit. We start at terminal $A$.

If we trace the wire originating from $A$, we see it bypasses the first resistor and connects directly to the junction between the second $2R$ resistor and the $R$ resistor. Because this entire path is just a plain wire, that junction is at the exact same potential as $A$. We can effectively label both ends of this wire as "Node $A$".

Now, let's do the same for terminal $B$. Tracing the wire backward from $B$, we find it connects to the junction between the first $2R$ resistor and the second $2R$ resistor. Again, this entire path is equipotential, so we label this junction as "Node $B$".

With our nodes clearly identified, let's re-examine how each resistor is connected: 1. The first $2R$ resistor is connected between the original Node $A$ and the newly identified Node $B$. 2. The second $2R$ resistor is connected between Node $B$ and Node $A$. 3. The final $R$ resistor is connected between Node $A$ and the original Node $B$.

Do you see the pattern? Every single resistor in this circuit is connected directly across the exact same two points: Node $A$ and Node $B$.

By definition, when components are connected across the same potential difference, they are in a parallel combination. The confusing web was just a disguise for a simple parallel circuit!

Now that we know the resistors are in parallel, we can easily find the equivalent resistance $R_{eq}$ using the standard parallel formula:

Substituting our specific resistance values into the equation:

Let's simplify the expression by adding the first two terms:

Now, add this result to the third term:

To find the final equivalent resistance, we simply take the reciprocal of both sides:

And there we have it! By systematically tracing the equipotential nodes, we transformed a daunting problem into a straightforward calculation, arriving at the elegant result of $R/2$.