Induction heating is a commonly used method in material processing where the

temperature in a piece of conductive material is raised by the eddy current induced by the

electromagnetic field.

Consider a metal plate of $2$ cm in thickness placed in an induction coil. The volumetric

heat generation $(Q_{\text{g}\text{e}\text{n}})$ is uniform, $1000\frac{W}{m^3}.$ The temperature at both sides of the plate is

measured $200C$ using an Infrared camera. The plate has a thermal conductivity of $200$

$\frac{W}{mC}.$ Solve the heat conduction equation and determine the temperature distribution

$T(x)$ at steady state.

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}})+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{y}})+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{z}})+q^{}=c\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}$