Question

The two ends of the uniform rod of thermal conductivity $ K $ are maintained at different but constant temperatures. The temperature gradient at any point on the rod is $ \dfrac{d \theta}{d l} $ (equal to the difference in temperature per unit length). The heat flow per unit time per unit cross section of the rod is $ l $ .
(A) $ \dfrac{d \theta}{d l} $ is the same for all points on the rod.
(B) $ l $ will decrease as we move from higher to lower temperature.
(C) $ l=k\cdot \dfrac{d\theta }{dl} $
(D) All the above options are correct.

Answer 1

Hint
We should know that thermal conductivity is defined as the ability of a given material to conduct or we can say transfer heat. It is generally denoted by the symbol k. The reciprocal of thermal conductivity is known as thermal resistivity. We can also say that thermal conductivity is defined as the rate at which heat is transferred by conduction through a unit cross-sectional area of a material, when a temperature gradient exists perpendicular to the area. Based on this concept we have to solve this question.

Complete step by step answer
The heat flow can be expressed with the equation,
 $ \dot{Q}=K A \dfrac{d \theta}{d l} $
Where, K is the thermal conductivity and $ \dfrac{d \theta}{d l} $ is the temperature gradient at any point of the rod which is also equivalent to the difference in temperature per unit length.
Given that $ A=1 $ (equivalent to unit cross section) and heat flow $ =l $ ;
 $ l=K \dfrac{d \theta}{d l} $
 $ \dfrac{d \theta}{d l}=\dfrac{l}{K}= $ constant
Being a multiple-choice solution,
We see that Option A and Option C satisfies the conditions while Option B does not. Option D is also incorrect since Option B is incorrect.

Therefore, the correct answers are Option (A) and Option (C).

Note
We should know that a temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. The temperature gradient is a dimensional quantity which is expressed in units of degrees, on a particular temperature scale, per unit length.
It should be known to us that the temperature gradient between places results in differences in air pressure and ultimately, wind. With the increase in the wind speed, the temperature difference is greater. Winds are known to re-distribute energy around the world.