Question

Two ends of a uniform conductor of length / are attemperature 10°C and 100°C respectively. Thetemperature at point P at distance from the hotend, as shown in the figure, is (assuming heattransfers by conduction only)

Answer 1

Given:

Two ends of a uniform conductor of length are at temperature 10°C and 100°C respectively.

To find:

Temperature at point P :

Calculation:

Let length of conductor be L and temperature at point P be $\theta$

Let point P be situated at a distance of "x" from hot end .

Since Heat Current doesn't accumulate at a point, we can say :

$\sf{ \therefore \: H1 = H2}$

$\sf{ = > \: \dfrac{100 - \theta}{ \{ \frac{x}{k(area)} \} } = \dfrac{ \theta - 10}{ \{ \frac{L - x}{k(area)} \} }}$

$\sf{ = > \: \dfrac{100 - \theta}{ \{ \frac{x}{ \cancel{k(area)}} \} } = \dfrac{ \theta - 10}{ \{ \frac{L - x}{ \cancel{k(area)}} \} }}$

$\sf{ = > \: \dfrac{100 - \theta}{ \{ x \} } = \dfrac{ \theta - 10}{ \{ L - x\} }}$

$\sf{ = > \: \theta L = 100L - 100x + 10x}$

$\sf{ = > \: \theta L = 100L - 90x }$

$\sf{ = > \: \theta = \dfrac{100L - 90x}{L} }$

So , temperature at point P is :

$\boxed{ \red{ \sf{ \: \theta = \dfrac{100L - 90x}{L} }}}$