Animated Solution for Physics - Current Electricity: Five equal resistances are connected in a network as shown in figure. The net resistance between the points $A$ and $B$ is

When you first look at this circuit, it might seem like a tangled web of resistors. We have five identical resistors, each with a resistance of $R$, connected in a somewhat triangular fashion. Our primary objective is to find the equivalent resistance, $R_{eq}$, between the input terminal $A$ and the output terminal $B$.

The key to solving complex circuits is often to redraw them into a topology that we easily recognize. Let's trace the paths. The current enters at terminal $A$ and immediately splits into two paths: one going to node $D$ and the other to node $E$.

Now, look at the output side. Terminal $B$ is directly connected to node $C$. This means node $C$ and terminal $B$ are electrically the exact same point. From nodes $D$ and $E$, the current flows through resistors to reach this common output point $C$ (or $B$). Finally, there is a resistor connecting node $D$ directly to node $E$.

Wheatstone Bridge

When we stretch this circuit out based on our tracing, a beautiful and familiar shape emerges: the Wheatstone Bridge.

We have four outer arms: $R_1$ (between $A$ and $D$), $R_2$ (between $D$ and $B$), $R_3$ (between $A$ and $E$), and $R_4$ (between $E$ and $B$). The fifth resistor acts as the central "bridge" connecting nodes $D$ and $E$.

Whenever you spot a Wheatstone bridge, your immediate reflex should be to check if it is balanced. The balance condition dictates that the ratio of the resistances in the opposite arms must be equal:

Let's substitute our given values. Since all five resistors are identical, $R_1=R_2=R_3=R_4=R$.

The ratios are perfectly equal. Our bridge is balanced!

Why do we care if it's balanced? Because in a balanced Wheatstone bridge, the nodes connected by the central resistor (nodes $D$ and $E$) are at the exact same electrical potential ($V_D=V_E$).

Since there is no potential difference across the central resistor, no current will flow through it. It becomes electrically "dead" weight, and we can completely remove it from our circuit analysis.

Once the central resistor is gone, the circuit simplifies drastically. The current flowing through the top branch goes straight from $R_1$ into $R_2$. They are in series.

Similarly, the bottom branch resistors $R_3$ and $R_4$ are in series.

Now, we are left with two branches, each of resistance $2R$, connected in parallel between terminals $A$ and $B$. The equivalent resistance of two equal parallel resistors is simply half of their individual value:

And there we have it! The equivalent resistance of the entire network is simply $R$.