Find the equivalent resistance between the terminals A and B in the network.

Steps to Solve

1. Identify the resistors in the circuit

   The circuit has four resistors with values R, 2R, 2R, and R arranged in a specific configuration between terminals A and B.

2. Simplify the resistors in series and parallel

   Group the resistors to simplify the calculation:

   • The two 2R resistors are in series with each other and can be combined: $$ R_{series} = 2R + 2R = 4R $$

3. Combine series and parallel resistors between A and B

   Now we have:

   • A resistor of R (between A and the point between series resistors)
   • A parallel combination of R and 4R from the previous step:

   The formula for equivalent resistance in parallel ($R_{eq}$) is: $$ \frac{1}{R_{eq}} = \frac{1}{R} + \frac{1}{4R} $$

4. Calculate the equivalent resistance

   To find $R_{eq}$, we rearrange the previous equation: $$ R_{eq} = \frac{1}{\frac{1}{R} + \frac{1}{4R}} $$ Combine: $$ R_{eq} = \frac{1}{\frac{4 + 1}{4R}} = \frac{4R}{5} $$

5. Combine with the remaining resistor

   Now, we have a resistor of $\frac{4R}{5}$ in series with the remaining R: $$ R_{total} = R + R_{eq} = R + \frac{4R}{5} = \frac{5R + 4R}{5} = \frac{9R}{5} $$