For example, Maxwellâs equations for an isotropic, time-invariant and homogeneous medium without electric or magnetic losses are given by:
:<math>\dfrac{\delta \mathbf{H}}{\delta t} = -\dfrac{1}{\mu} \Delta \times \mathbf{E}</math>
:<math>\dfrac{\delta \mathbf{E}}{\delta t} = -\dfrac{1}{\epsilon} \Delta \times \mathbf{H}</math>
where '''E''' and '''H''' are the electric and magnetic fields, respectively, ε is the permittivity and μ is the permeability. Both time- and space-derivatives are approximated with central finite differences. This results in six differential equations, one for each field component. For the field components in x-direction, the field equations result in:
:<math> H_x^{n+\frac{1}{2}} (i,j,k) = H_x^{n-\frac{1}{2}} (i,j,k) + \frac{\Delta t}{\mu (i,j,k)} \left[ \frac{E_{y}^{n}(i,j,k+1) - E_{y}^{n}(i,j,k)}{\Delta z} - \frac{E_{z}^{n} (i,j+1,k)-E_{z}^{n} (i,j,k)}{\Delta y} \right] </math>
:<math> {E_{x}^{n+1} (i,j,k) = E_{x}^{n} (i,j,k)} + \frac{\Delta t}{\epsilon (i,j,k)} \left[ \frac{H_{z}^{n+\frac{1}{2} } (i,j,k) - H_{z}^{n+\frac{1}{2} } (i,j-1,k)}{\Delta y} - \frac{H_{y}^{n+\frac{1}{2} } (i,j,k) - H_{y}^{n+\frac{1}{2} } (i,j,k-1)}{\Delta z} \right] </math>
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