Two Rods of equal length have same temperature difference between their ends. If their thermal conductivities are K1 and K2 and their areas of cross section are A1 and A2 respectively, Find the condition that will ensure equal rates of flow of heat through both the rods.
K1/k2=A2/A1
Correct answer
To ensure equal rates of flow of heat through both rods, we need to arrange the thermal conductivities and areas of cross section in such a way that the thermal conductivity multiplied by the area of cross section is the same for both rods. Mathematically, it can be written as:
K1 * A1 = K2 * A2
This equation represents the condition that will ensure the equal rates of flow of heat through both the rods.
To find the condition that ensures equal rates of flow of heat through both rods, we can use the formula for heat conduction:
Q = (k * A * ΔT) / L
where Q is the rate of flow of heat, k is the thermal conductivity, A is the cross-sectional area, ΔT is the temperature difference, and L is the length of the rod.
Let's assume that the temperature difference between the ends of each rod is ΔT.
For the first rod:
Q1 = (K1 * A1 * ΔT) / L
For the second rod:
Q2 = (K2 * A2 * ΔT) / L
Now, we need to find the condition that will make Q1 equal to Q2.
Setting Q1 equal to Q2:
(K1 * A1 * ΔT) / L = (K2 * A2 * ΔT) / L
Since the lengths (L) and temperature differences (ΔT) are the same for both rods, we can eliminate them from the equation:
(K1 * A1) = (K2 * A2)
This equation shows that the product of the thermal conductivity and the cross-sectional area of each rod must be equal in order to ensure equal rates of flow of heat through both rods.