If `sigma_(1), sigma_(2)` and `sigma_(3)` are the conductances of three conductors, then their equivalent conductance, when they are joined in series, will be-

Options

A. `sigma_(1) + sigma_(2) + sigma_(3)`
B. `1/sigma_(1) + 1/sigma_(2) + 1/sigma_(3)`
C. `(sigma_(1) sigma_(2) sigma_(3))/(sigma_(1) + sigma_(2) + sigma_(3))`
D. None of these

Explanation

To find the equivalent conductance of three conductors connected in series, we can follow these steps:

### Step 1: Understand the relationship between conductance and resistance
Conductance (σ) is the reciprocal of resistance (R). Therefore, we can express this relationship as:
\[ R = \frac{1}{\sigma} \]

### Step 2: Write the expression for total resistance in series
When resistors (or conductors) are connected in series, the total resistance (R_eq) is the sum of the individual resistances:
\[ R_{eq} = R_1 + R_2 + R_3 \]

### Step 3: Express individual resistances in terms of conductance
Using the relationship from Step 1, we can express the individual resistances in terms of their conductances:
\[ R_1 = \frac{1}{\sigma_1}, \quad R_2 = \frac{1}{\sigma_2}, \quad R_3 = \frac{1}{\sigma_3} \]

### Step 4: Substitute the expressions for resistance into the total resistance formula
Now substituting the expressions for R_1, R_2, and R_3 into the total resistance equation:
\[ R_{eq} = \frac{1}{\sigma_1} + \frac{1}{\sigma_2} + \frac{1}{\sigma_3} \]

### Step 5: Find the equivalent conductance
The equivalent conductance (σ_eq) is the reciprocal of the equivalent resistance:
\[ \sigma_{eq} = \frac{1}{R_{eq}} \]
Substituting the expression for R_eq from Step 4:
\[ \sigma_{eq} = \frac{1}{\left(\frac{1}{\sigma_1} + \frac{1}{\sigma_2} + \frac{1}{\sigma_3}\right)} \]

### Step 6: Simplify the expression
To simplify this expression, we can find a common denominator:
\[ \sigma_{eq} = \frac{\sigma_1 \sigma_2 \sigma_3}{\sigma_2 \sigma_3 + \sigma_1 \sigma_3 + \sigma_1 \sigma_2} \]

### Final Result
Thus, the equivalent conductance when the conductors are joined in series is given by:
\[ \sigma_{eq} = \frac{\sigma_1 \sigma_2 \sigma_3}{\sigma_2 \sigma_3 + \sigma_1 \sigma_3 + \sigma_1 \sigma_2} \]

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