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The thermo emf E (in volts) of a certain...

The thermo emf E (in volts) of a certain thermocouple is found to vary with Q (in C) acoording to eqution `(E=20Q-Q^2/(20))`, where Q is tempereature of the hot function, the cold function being kept at `0^@C`. Then the neutral temperature of the thermocouple is

A

`300^@C`

B

`400^@C`

C

`100^@C`

D

`200^@C`

Text Solution

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The correct Answer is:
To find the neutral temperature of the thermocouple given the equation for thermo EMF \( E \) as a function of temperature \( Q \), we can follow these steps: ### Step 1: Write down the equation for thermo EMF The equation given is: \[ E = 20Q - \frac{Q^2}{20} \] ### Step 2: Differentiate the equation with respect to \( Q \) To find the neutral temperature, we need to find the maximum value of \( E \). This occurs when the derivative of \( E \) with respect to \( Q \) is zero: \[ \frac{dE}{dQ} = 0 \] Differentiating \( E \): \[ \frac{dE}{dQ} = 20 - \frac{2Q}{20} \] This simplifies to: \[ \frac{dE}{dQ} = 20 - \frac{Q}{10} \] ### Step 3: Set the derivative equal to zero Now, we set the derivative equal to zero to find the value of \( Q \) at which \( E \) is maximum: \[ 20 - \frac{Q}{10} = 0 \] ### Step 4: Solve for \( Q \) Rearranging the equation gives: \[ \frac{Q}{10} = 20 \] Multiplying both sides by 10: \[ Q = 200 \] ### Step 5: State the neutral temperature The neutral temperature of the thermocouple is therefore: \[ \text{Neutral Temperature} = 200 \, ^\circ C \] ### Final Answer The neutral temperature of the thermocouple is \( 200 \, ^\circ C \). ---
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