Question

The equivalent resistance between diagonally opposite corners A and B is:

A. $R$
B. $R/3$
C. $2R/3$
D. $4R/3$

Answer 1

Use the formulae for the equivalent resistance for the resistors connected in series arrangement and parallel arrangement. Observe the given circuit and draw the equivalent circuit diagram of the given circuit. From this equivalent circuit diagram, determine the net resistance between the terminals A and B using the formula for equivalent resistance of series and parallel combination.

Formulae used:
The equivalent resistance ${R_{eq}}$ between the two resistors of resistances ${R_1}$ and ${R_2}$ connected in series is
${R_{eq}} = {R_1} + {R_2}$ …… (1)
The equivalent resistance ${R_{eq}}$ between the three resistors of resistances ${R_1}$, ${R_2}$ and ${R_3}$ connected in series is
$\dfrac{1}{{{R_{eq}}}} = \dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}} + \dfrac{1}{{{R_3}}}$ …… (2)

Complete step by step answer:
We have given a circuit diagram in which the resistors of resistance $R$ are connected.From the figure, we can conclude that the current flows through all the resistors is the same.But from the given circuit diagram, we can conclude that the potential difference for the resistors connected in the vertical arm CD of the given circuit is the same. So, there is no current flowing through the resistors in the vertical arm CD of the circuit diagram. Hence, we can neglect the resistors connected in the vertical arm CD.

Therefore, the equivalent circuit diagram for the given circuit can be drawn as follows:

In the above circuit diagram, three rows in which two resistors are connected in series are in parallel. The equivalent resistance $R'$ of the two resistors connected in series is
$R' = R + R$
$\Rightarrow R' = 2R$
Hence, the equivalent resistance of the two resistors connected in series is $2R$.There are three such rows. Let us determine the equivalent resistance ${R_{eq}}$ of the three rows connected in parallel.
Substitute $2R$ for ${R_1}$, ${R_2}$ and ${R_3}$ in equation (2).
$\dfrac{1}{{{R_{eq}}}} = \dfrac{1}{{2R}} + \dfrac{1}{{2R}} + \dfrac{1}{{2R}}$
$\Rightarrow \dfrac{1}{{{R_{eq}}}} = \dfrac{3}{{2R}}$
$\therefore {R_{eq}} = \dfrac{{2R}}{3}$
Therefore, the equivalent resistance between the diagonally opposite terminals is $\dfrac{{2R}}{3}$.

Hence, the correct option is C.

Note: The students should be careful while drawing the equivalent circuit diagram of the given circuit. While drawing this equivalent circuit diagram, the students should check the potential difference across each resistor connected in the circuit. The potential difference across the resistors in the vertical arm CD of the given circuit diagram is zero. Hence, we should eliminate this branch while drawing this circuit diagram.