Name: Seebeck Effect
Uploaded: 2019-07-19
Duration: 3 min 49 s
Description: The Seebeck effect is a phenomenon in which a temperature difference between two dissimilar electrical conductors produces a voltage difference between two s...

Question

State and prove the principle of least heat or the minimum heating effect of current.

Accepted Answer

When a current divides into a number of branches in parallel, it does so in such a way that the total heat produced is a minimum.

We see that a current $I$ branches into two resistors of resistances $R_{1}$ and $R_{2}$ in parallel, the currents being $I_{1}$ and $I_{2} = (I - I_{1})$ respectively. If $Q$ is the heat produced per second in the two wires, then

$Q = {I_{1}}^{2} R_{1} + {I_{2}}^{2} R_{2} = {I_{1}}^{2} R_{1} + {(I - I_{1})}^{2} R_{2}$ .

Now, let us see for which value of $I_{1}$ , $Q$ is minimum. The necessary condition is $\frac{d Q}{d I_{1}} = 0$ .

Now,

$\frac{d Q}{d I_{1}} = 2 I_{1} R_{1} + 2 (I - I_{1}) (- 1) R_{2} = 0 I_{1} R_{1} = (I - I_{1}) R_{2} = I_{2} R_{2} \frac{I_{1}}{I_{2}} = \frac{R_{2}}{R_{1}}$

Thus we find that the current is divided in the inverse ratio of the resistances. This is exactly what happens. If we find the second derivation, $\frac{d^{2} Q}{d I_{1}} = 2 (R_{1} + R_{2}) = + ve .$ So it is confirmed that the above is the condition for minimum heat. Also, the heating effect of an electric current depends on the resistance of the conductor, the time for which the current flows and the amount of current.

State and prove the principle of least heat or the minimum heating effect of current.

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