Question

If ${\sigma _1}$ , ${\sigma _2}$ , and ${\sigma _3}$ the conductances of three conductors then their equivalent conductance when they are joined in series will be
(A) ${\sigma _1} + {\sigma _2} + {\sigma _3}$
(B) $\dfrac{1}{{{\sigma _1}}} + \dfrac{1}{{{\sigma _2}}} + \dfrac{1}{{{\sigma _3}}}$
(C) $\dfrac{{{\sigma _1}{\sigma _2}{\sigma _3}}}{{{\sigma _1} + {\sigma _2} + {\sigma _3}}}$
(D) None of these

Answer 1

To solve this question, we need to find out the resistance of each conductor in terms of the value of the conductance given. Then, find out the value of the equivalent resistance of the combination of these conductors. Finally convert the equivalent resistance back to the equivalent resistance to get the final answer.
Formula Used: The formula used in solving this question is
- ${R_s} = {R_1} + {R_2} + {R_3}$ , where ${R_s}$ is the series equivalent resistance of the resistances ${R_1}$ , ${R_2}$ , and ${R_3}$
- $R = \dfrac{1}{\sigma }$ , where $\sigma$ is the conductance and $R$ is the resistance.

Complete step by step answer:
Let the equivalent conductance be ${\sigma _s}$
It is given that the conductances of the conductors are ${\sigma _1}$ , ${\sigma _2}$ , and ${\sigma _3}$
We know that the resistance is related to the conductance is given by
$R = \dfrac{1}{\sigma }$
So, the resistance of the first conductor is
${R_1} = \dfrac{1}{{{\sigma _1}}}$ …………………..(i)
The resistance of the second conductor is
${R_2} = \dfrac{1}{{{\sigma _2}}}$ …………………..(ii)
The resistance of the third conductor is
${R_3} = \dfrac{1}{{{\sigma _3}}}$ …………………..(iii)
And the equivalent resistance is
${R_s} = \dfrac{1}{{{\sigma _s}}}$ …………………..(iv)
According to the question, the conductors are arranged in series combination. So, the equivalent resistance is given by
${R_s} = {R_1} + {R_2} + {R_3}$
Putting (i), (ii), (iii) and (iv), we get
$\dfrac{1}{{{\sigma _s}}} = \dfrac{1}{{{\sigma _1}}} + \dfrac{1}{{{\sigma _2}}} + \dfrac{1}{{{\sigma _3}}}$
Now, taking the LCM, we have
$\dfrac{1}{{{\sigma _s}}} = \dfrac{{{\sigma _2}{\sigma _3} + {\sigma _1}{\sigma _3} + {\sigma _1}{\sigma _2}}}{{{\sigma _1}{\sigma _2}{\sigma _3}}}$
Finally, taking the reciprocal, we get
${\sigma _s} = \dfrac{{{\sigma _1}{\sigma _2}{\sigma _3}}}{{{\sigma _2}{\sigma _3} + {\sigma _1}{\sigma _3} + {\sigma _1}{\sigma _2}}}$
We see that we don’t have this value of the equivalent conductance in any of the options given.
Therefore, the options A, B and C are incorrect.
Hence, the correct answer is option (D), none of these.

Note:
Do not choose option A as the correct answer, as this is the value of the reciprocal of the equivalent conductance. This value is coming in between the steps of our evaluation, so this mistake can be committed. To avoid this mistake, do not convert the equivalent resistance ${R_s}$ in terms of the equivalent conductance ${\sigma _s}$ in the beginning. Calculate the equivalent resistance ${R_s}$ in terms of the values of the conductances given. Then, you will remember to take the reciprocal in the end.