Heat Equation for No Heat Loss at x=a

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The discussion centers on the heat equation for two rods, L1 and L2, joined at x=a, where no heat energy is lost at the junction. The key conclusion is that the condition for no energy loss can be represented by equating the rate of heat transfer from rod L1 to the rate of heat transfer into rod L2. This approach utilizes the principles of thermal conductivity and continuity of temperature at the junction.

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The question :Two rods L1 and L2 of different materials( hence different thermal conductivities) and different cross-sectional areas,are joined at x=a. The temperature is continuous,
And NO HEAT ENERGY IS LOST AT a, so all heat energy that flows from L1 flows into L2.

? What equation represents the condition that no energy is lost at a ?

My thoughts : Equate the rate at which heat leaves L1 to the rate at which heat enters L2. ?
 
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