The problem is asking for the ratio of the equivalent resistance between points $A$ and $B$ to the equivalent resistance between points $B$ and $C$. Let's analyze and solve it step by step.

Step 1: Identify the Circuit Configuration

The circuit is a Wheatstone bridge with all resistors equal to $R$. The resistances are arranged as follows:

• Four resistors form a diamond shape with $R$ each.
• One resistor is in the middle (also $R$), connecting the two midpoints of the bridge.

Since all resistances are equal, the bridge is balanced, meaning no current flows through the middle resistor. This allows us to ignore the middle resistor when calculating equivalent resistance.

Step 2: Find the Equivalent Resistance Between $A$ and $B$

• The two top resistors (between $A$ and $B$) are in parallel.
• Similarly, the two bottom resistors (between $A$ and $B$) are in parallel.

Each parallel combination of two resistors of resistance $R$ results in: $R_{\text{parallel}} = \frac{R \cdot R}{R + R} = \frac{R}{2}$ Now, these two parallel resistances are in series, so the total equivalent resistance between $A$ and $B$ is: $R_{AB} = \frac{R}{2} + \frac{R}{2} = R$

Step 3: Find the Equivalent Resistance Between $B$ and $C$

• The two resistors between $B$ and $C$ are in parallel, giving: $R_{BC} = \frac{R \cdot R}{R + R} = \frac{R}{2}$

Step 4: Compute the Ratio

$\frac{R_{AB}}{R_{BC}} = \frac{R}{\frac{R}{2}} = 2$

Final Answer:

The equivalent resistance between points $A$ and $B$ is twice the equivalent resistance between points $B$ and $C$.
$\boxed{2}$

Would you like a more detailed breakdown or further clarifications?

Related Questions:

1. How does the concept of a balanced Wheatstone bridge help simplify circuit analysis?
2. What happens to the equivalent resistance if one of the resistors is different from $R$?
3. How would the equivalent resistance change if the middle resistor were removed?
4. What is the effective resistance between any two adjacent nodes in a square resistor network?
5. Can the star-delta (Y-Δ) transformation be used to simplify this circuit?

Tip:

In circuit analysis, always check for symmetry and balanced conditions to simplify calculations efficiently.