Question

A uniform wire of resistance R is cut into three equal lengths. One of these is formed into a circle and connected between the other two (Fig. E26.1). What is the resistance between the opposite ends a and b?

Short answer

The resistance between the opposite ends a and b is 3R/4$\Omega$

Step-by-step solution

Understanding of the circuit

The circle has a total resistance of R/ 3. Also, it acts as two resistors connected in parallel as shown in the figure below where each one of them is R_C1 and R_C2. These two resistors are equal to each other so the total resistance of the circle is split into these resistors and each one has a resistor equal to R/6. Where the circle has resistance R/3.

Calculation of the equivalent resistance

Calculate the equivalent resistance of the entire wire. R_C1and R_C2 are in parallel and to calculate their combination:

$$\begin{gathered} R_{ceq}=\frac{R_{c1}{R}_{c2}}{R_{c1}+R_{c2}} \\ =\frac{\left(R/6\right)\left(R/6\right)}{R/6+R/6} \\ =R/12 \end{gathered}$$

Calculation of resistance between a and b

This combination for the circle is in series with the two other resistors R₁ and R₃ as shown in the figure.

$R_{eq}=R_1+R_{Ceq}+R_3$

The total equivalent resistance of the wire

$$\begin{gathered} R_{eq}=R_1+R_{Ceq}+R_3 \\ =\frac{R}{3}+\frac{R}{12}+\frac{R}{3} \\ =\frac{3R}{4} \end{gathered}$$

The resistance between the opposite ends a and b is 3R/4$\Omega$.