Question

In the figure shown, each resistance is R.

Do the conditions of Table 1 match the values of Table 2

state true or false?

Table 1	Table 2
(a) Resistance between a and b	(p)$\frac{R}{2}$
(b) Resistance between a and c	(q)$\frac{5}{8}R$
(c) Resistance between b and d	(r) R

• A. True
• B. False

Answer 1

The correct option is B False

Given: A circuit with each resistance equal to R

To find the resistance between various points

Solution:

(a) Resistance between a and b ( refer fig(a))

In this case resistance 1 and 2 are in series. Now resistance 1' and 3 are in parallel. Now resistance 1'' and 4 are in series. In the last part we can see that we obtain a simplified circuit where resistance 1''' and 5 are in parallel.

Let the resistance between a and b be R'

$\frac{1}{R^{\prime }}=\frac{1}{\frac{5R}{3}}+\frac{1}{R}⟹\frac{1}{R^{\prime }}=\frac{3}{5R}+\frac{1}{R}⟹\frac{1}{R^{\prime }}=\frac{3+5}{5R}=\frac{8}{5R}$

Hence the resistance between a and b is $\frac{5}{8}R$

(b) Resistance between a and c (refer fig(b))

In this case resistance 1 and 2 are in series and resistance 3 and 4 are also in series. In the last part we can see that we obtain a simplified circuit where resistance 1', 2' and 5 are in parallel.

Let the resistance between a and b be R''

$\frac{1}{R^{\prime \prime }}=\frac{1}{2R}+\frac{1}{R}+\frac{1}{2R}⟹\frac{1}{R^{\prime \prime }}=\frac{1+2+1}{2R}=\frac{4}{2R}=\frac{2}{R}$

Hence the resistance between a and b is $\frac{R}{2}$

(c) Resistance between b and d(refer fig(c))

In this case no current passes through resistance 3, as the current passing through point b should be equal to current received at point d. If some current passes through resistance 3 then the current passed is not equal to the current received. And so we obtain second part where I have removed resistance 3 from the circuit. Now resistance 1 and 4 are in series and similarly resistance 2 and 5 are in series. In the last part we can see that we obtain a simplified circuit where resistance 1' and 2' are in parallel.

Let the resistance between b and d be R'''

$\frac{1}{R^{\prime \prime \prime }}=\frac{1}{2R}+\frac{1}{2R}⟹\frac{1}{R^{\prime \prime \prime }}=\frac{1+1}{2R}=\frac{2}{2R}=\frac{1}{R}$

Hence the resistance between b and d is $R$