Consider a uniform rod of material whose temperature varies only along its length, in the x direction. By considering the heat flowing from both directions into a small segment of length Δx, derive the heat equation,

where K= κt/cρ, c is the specific heat of the material, and ρ is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that K is independent of temperature, show that a solution of the heat equation is

where T0 is a constant background temperature and A is any constant. Sketch (or use a computer to plot) this solution as a function of x, for several values of t. Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.

Solution 62P

To solve this question, we shall have to consider the Fourier heat conduction law. The mathematical equation for Fourier heat conduction law is written as …..(1)

Here, amount of heat

time taken for the heat to flow

thermal conductivity

area of cross-section

change in temperature

change in length

Step 1</p>

We are given in the question that heat flows from both directions of the rod.

Using equation (1),

For heat coming from one direction,

…..(2)

For heat conduction from the other direction.

…..(3)

Therefore, the net heat flow can be subtracting equation (3) from equation (2),

…..(2), where …..(4)