The temperatures of the junctions of a bismuth-silver thermocouple are maintained at `0^@ C` and `0.001^@ C`. Find the thermo-emf (Seebeck emf) developed. For bismuth-silver, `a = - 46 xx 10^(-6) V^@ C^(-1) and b= -0.48 xx 10^(-6) V^@ V^@ C^(-2)`.

Explanation

To find the thermo-emf (Seebeck emf) developed in a bismuth-silver thermocouple when the temperatures of the junctions are maintained at \(0^\circ C\) and \(0.001^\circ C\), we can use the formula for the Seebeck effect:

\[ E = a \theta + \frac{1}{2} b \theta^2 \]

Where:
- \(E\) is the thermo-emf,
- \(a\) is the first coefficient of thermoelectric power,
- \(b\) is the second coefficient of thermoelectric power,
- \(\theta\) is the temperature difference in degrees Celsius.

### Step 1: Calculate the Temperature Difference
The temperature difference \(\theta\) between the two junctions is:
\[ \theta = 0.001^\circ C - 0^\circ C = 0.001^\circ C \]

### Step 2: Substitute Values into the Formula
Given:
- \(a = -46 \times 10^{-6} \, V/^\circ C\)
- \(b = -0.48 \times 10^{-6} \, V/^\circ C^2\)

Substituting the values into the Seebeck effect formula:
\[ E = (-46 \times 10^{-6}) \times (0.001) + \frac{1}{2} \times (-0.48 \times 10^{-6}) \times (0.001)^2 \]

### Step 3: Calculate Each Term
1. Calculate the first term:
\[ E_1 = -46 \times 10^{-6} \times 0.001 = -46 \times 10^{-9} \, V \]

2. Calculate the second term:
\[ E_2 = \frac{1}{2} \times (-0.48 \times 10^{-6}) \times (0.001)^2 = \frac{1}{2} \times (-0.48 \times 10^{-6}) \times (0.000001) = -0.24 \times 10^{-12} \, V \]

### Step 4: Combine the Results
Now, we combine both terms:
\[ E = E_1 + E_2 = -46 \times 10^{-9} + (-0.24 \times 10^{-12}) \]
Since \(-0.24 \times 10^{-12}\) is much smaller than \(-46 \times 10^{-9}\), we can approximate:
\[ E \approx -46 \times 10^{-9} \, V \]
This can be expressed as:
\[ E \approx -4.6 \times 10^{-8} \, V \]

### Final Answer
The thermo-emf developed is approximately:
\[ E \approx -4.6 \times 10^{-8} \, V \]