`sigma_(1), sigma_(2), sigma_(3)` are the conductances of three conductors. What will be their equivalent conductance when they are connected, (i) in series (ii) in parallel.

Options

A. `sigma_1+sigma_2 +sigma_3`
B. `1/sigma_1+2/sigma_2 +1/sigma_3`
C. `(sigma_1sigma_2sigma_3)/(sigma_1+sigma_2+sigma_3)`
D. none of these

Answer

To find the equivalent conductance of three conductors with conductances \( \sigma_1, \sigma_2, \sigma_3 \) when connected in series and in parallel, we can follow these steps:

### Step-by-step Solution:

**(i) Equivalent Conductance in Series:**

1. **Understand the relationship between conductance and resistance:**
Conductance \( \sigma \) is the reciprocal of resistance \( R \):
\[ \sigma = \frac{1}{R} \quad \text{or} \quad R = \frac{1}{\sigma} \]

2. **Write the expression for equivalent resistance in series:**
For resistors in series, the equivalent resistance \( R_{\text{eq}} \) is given by:
\[ R_{\text{eq}} = R_1 + R_2 + R_3 \]

3. **Substitute the conductance values:**
Using the relationship \( R_i = \frac{1}{\sigma_i} \):
\[ R_{\text{eq}} = \frac{1}{\sigma_1} + \frac{1}{\sigma_2} + \frac{1}{\sigma_3} \]

4. **Take the reciprocal to find equivalent conductance:**
The equivalent conductance \( \sigma_{\text{eq}} \) is given by:
\[ \sigma_{\text{eq}} = \frac{1}{R_{\text{eq}}} = \frac{1}{\left(\frac{1}{\sigma_1} + \frac{1}{\sigma_2} + \frac{1}{\sigma_3}\right)} \]

5. **Simplify the expression:**
To simplify, find a common denominator:
\[ \sigma_{\text{eq}} = \frac{\sigma_1 \sigma_2 \sigma_3}{\sigma_2 \sigma_3 + \sigma_3 \sigma_1 + \sigma_1 \sigma_2} \]

**(ii) Equivalent Conductance in Parallel:**

1. **Write the expression for equivalent resistance in parallel:**
For resistors in parallel, the equivalent resistance \( R_{\text{eq}} \) is given by:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]

2. **Substitute the conductance values:**
Using the relationship \( R_i = \frac{1}{\sigma_i} \):
\[ \frac{1}{R_{\text{eq}}} = \sigma_1 + \sigma_2 + \sigma_3 \]

3. **Find the equivalent conductance:**
The equivalent conductance \( \sigma_{\text{eq}} \) is simply:
\[ \sigma_{\text{eq}} = \sigma_1 + \sigma_2 + \sigma_3 \]

### Final Results:
- **Equivalent Conductance in Series:**
\[ \sigma_{\text{eq (series)}} = \frac{\sigma_1 \sigma_2 \sigma_3}{\sigma_2 \sigma_3 + \sigma_3 \sigma_1 + \sigma_1 \sigma_2} \]

- **Equivalent Conductance in Parallel:**
\[ \sigma_{\text{eq (parallel)}} = \sigma_1 + \sigma_2 + \sigma_3 \]