# What is temperature as a function of time in Fourier's Law?

Can we find temperature as a function of time using Fourier's Law of thermal conduction?

For example:

If we have two square parallel plates of a given length, width, and distance apart with one plate hotter than the other, how do find the temperature of the two plates as a function of time?

Fourier's Law in 1D says:

$$q_x = -k\frac{dT}{dx}$$

I'm not sure how to integrate/manipulate this equation to get temperature with a time dependence. Any advice would be appreciated!

Fourier's law is just expressing the heat flux as a function of the gradient of temperature in a medium characterized by the heat conductivity $$k$$. What's the effect of such heat flux on the temperature of the material depends on something else which are not present in the Fourier's law: the effect of a heat transfer on the temperature rate of variation. Under the hypothesis that all the heat entering in a small volume of material goes into internal energy, one can obtain the Fourier equation of heat ( https://en.wikipedia.org/wiki/Heat_equation ), which in one dimension, and for a uniform sample can be written as $$\frac{\partial{T}}{\partial{t}}= \frac{k}{c_p \rho}\frac{\partial^2{T}}{\partial{x}^2}$$ where $$c_p$$ is the constant pressure specific heat and $$\rho$$ the mass density. This is a partial differential equations and methods are known to solve it.
The physical problem of finding the temperature field between two $$physical$$ plates at different temperature, one in front of the other, may require more than the Fourier law (for example, if between the two surfaces there would be a gas, convective motion could play an important role).
In order to obtain the temperature time- and space-dependences $$T(x,t)$$, you have to solve the heat conduction equation with some initial conditions $$T(x,0)=f(x)$$ and with some boundary conditions. The Fourier law is not sufficient for that.