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

The Power of Identifying Short Circuits

When you first look at a circuit diagram filled with multiple resistors and a web of cross-connecting wires, it's easy to feel overwhelmed. The key to unraveling these puzzles is to systematically trace the nodes and look for hidden simplifications. Let's walk through this interesting problem and see how a seemingly complex network collapses into a beautifully simple structure.

Spotting the Short Circuit

Our first task is to carefully examine the connections. Look closely at the $5\Omega$ resistor located on the bottom left of the original diagram. Notice the vertical wires connecting the nodes immediately before and after this resistor to the top wire.

Because the top wire is a continuous conductor with no components between those connection points, it acts as a single electrical node. This means the wire creates a direct, zero-resistance path around the $5\Omega$ resistor.

In physics, we know that current always seeks the path of least resistance. Given the choice between struggling through a $5\Omega$ resistor or breezing through a zero-resistance wire, all the current will choose the wire. The $5\Omega$ resistor is completely short-circuited and can be safely discarded from our analysis.

Once we remove the shorted component, the true topology of the circuit begins to reveal itself. By tracing the remaining nodes, we can redraw the circuit into a much more familiar form.

In our simplified, resolved circuit, we can immediately spot that the first two $2\Omega$ resistors are actually connected in parallel between the same two nodes. Let's calculate their equivalent resistance, which we'll call $R_1$:

Now, we replace that parallel pair with our new $1\Omega$ equivalent resistor. Looking at the top branch of our redrawn circuit, this $1\Omega$ resistor is now in series with the next $2\Omega$ resistor.

Adding them together gives us the total resistance for the entire top branch, which we'll call $R_2$:

Look at what we have left! The complex web has beautifully collapsed. We now have exactly three branches, and each branch has a resistance of exactly $3\Omega$. Furthermore, all three of these branches are connected directly in parallel between our main terminals $A$ and $B$.

Calculating the final equivalent resistance ($R_{eq}$) for three identical resistors in parallel is straightforward:

So, the equivalent resistance of the entire given circuit is $1\Omega$. By carefully identifying the short circuit right at the beginning, a daunting problem was transformed into a quick and satisfying calculation.