The e.m.f. in a thermoelectric circuit with one junction at `0^(@)C` and the other at `t^(@)C` is given by `E=At-Bt^(2)`. The neutral temperature is then

Options

A. `(A)/(B)`
B. `-(A)/(2B)`
C. `-(B)/(2A)`
D. `(A)/(2B)`

Answer

To find the neutral temperature in a thermoelectric circuit where the e.m.f. is given by the equation \( E = At - Bt^2 \), we can follow these steps:

### Step 1: Understand the concept of neutral temperature
The neutral temperature is defined as the temperature at which the thermoelectric e.m.f. does not change with respect to temperature. Mathematically, this is represented by the condition that the derivative of e.m.f. with respect to temperature is zero.

### Step 2: Differentiate the e.m.f. equation
We start with the e.m.f. equation:
\[ E = At - Bt^2 \]
Now, we differentiate \( E \) with respect to \( t \):
\[ \frac{dE}{dt} = A - 2Bt \]

### Step 3: Set the derivative to zero
To find the neutral temperature, we set the derivative equal to zero:
\[ A - 2Bt = 0 \]

### Step 4: Solve for \( t \)
Rearranging the equation gives:
\[ 2Bt = A \]
Now, we can solve for \( t \):
\[ t = \frac{A}{2B} \]

### Step 5: Conclusion
Thus, the neutral temperature \( T_n \) is given by:
\[ T_n = \frac{A}{2B} \]

### Final Answer
The neutral temperature is \( \frac{A}{2B} \).

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