two road of equl lenght have same tempeities are differentce between thir ends.if their thermal conductivities arek1 ANDK2 and their areas of cross section are A1 AND A2 respectively find the conditionthat will ensure equil ratess of flow of heat through both the rods
To ensure equal rates of flow of heat through both the rods, we need to equate the rates of heat transfer for both rods.
The rate of heat transfer through a rod is given by the formula:
Rate of heat transfer (Q) = (Thermal conductivity * Area of cross section * Temperature difference) / Length
Let's consider the first rod with thermal conductivity K1, area of cross section A1, and length L. The temperature difference across the rod is ΔT1.
So, the rate of heat transfer through the first rod can be written as:
Q1 = (K1 * A1 * ΔT1) / L
Similarly, for the second rod with thermal conductivity K2, area of cross section A2, and length L, the temperature difference across the rod is ΔT2.
The rate of heat transfer through the second rod can be written as:
Q2 = (K2 * A2 * ΔT2) / L
To ensure equal rates of flow of heat through both rods, we set Q1 equal to Q2:
(K1 * A1 * ΔT1) / L = (K2 * A2 * ΔT2) / L
Here, L is the length of both rods and can be canceled out from both sides of the equation.
Now, to simplify the equation, we divide both sides by ΔT1 and ΔT2:
(K1 * A1) / ΔT1 = (K2 * A2) / ΔT2
Now, rearranging the equation, we can find the condition that ensures equal rates of flow of heat through both rods:
(K1 * A1) / (K2 * A2) = ΔT1 / ΔT2
Therefore, the condition that will ensure equal rates of flow of heat through both rods is that the ratio of the product of thermal conductivity and area of cross section for the first rod to the product of thermal conductivity and area of cross section for the second rod should be equal to the ratio of the temperature difference across the first rod to the temperature difference across the second rod.