Maxwell's Equations

Let us now try to apply the same ideas to Maxwell's equations. In free space we have

$$\nabla\times\bi{E} = - \mu_0{\partial\bi{H}\over\partial t}$$ (2.38)

$$\nabla\times\bi{H} = + \epsilon_0{\partial\bi{E}\over\partial t}.$$ (2.39)

In order to reverse the derivation of these equations we consider space to be divided into cubes as before. For (2.6.3a) we integrate over a face of the cube and apply Stokes' theorem

$$\oint_S\nabla\times\bi{E}\cdot{\bf d}\bi{S} = \int\bi{E}\cdot{\bf d}\bi{l}$$ (2.40)

$$= -{\partial\over\partial t}\oint_S\mu_0\bi{H}\cdot{\bf d}\bi{S}$$ (2.41)

and the integral of the electric field, $\bi{E}$, round the edges of the face is equal to minus the rate of change of the magnetic flux through the face, i.e. Faraday's law. Here we can associate the electric field, $\bi{E}$ with the edges and the magnetic field with the face of the cube. In the case of the diffusion equation we had to think in terms of the total charge in a cube instead of the density. Now we replace the electric field with the integral of the field along a line and the magnetic field with the flux through a face. Note also that we can analyse (2.6.3b) in a similar way to obtain a representation of Ampère's law.