Maxwell's Equations

Here, , , , and represent the

(5) |

is the

(6) |

is the

Maxwell's equations are linear in nature. In other words, if and , where is an arbitrary (spatial and temporal) constant, then it is clear from Equations (1)-(4) that and . The linearity of Maxwell's equations accounts for the well-known fact that the electric fields generated by point charges, as well as the magnetic fields generated by line currents, are superposable.

Taking the divergence of Equation (4), and combining the resulting expression with Equation (1), we obtain

In integral form, making use of the divergence theorem, this equation becomes

(8) |

where is a fixed volume bounded by a surface . The volume integral represents the net electric charge contained within the volume, whereas the surface integral represents the outward flux of charge across the bounding surface. The previous equation, which states that the net rate of change of the charge contained within the volume is equal to minus the net flux of charge across the bounding surface , is clearly a statement of the

As is well known, a point electric charge moving with velocity in the presence of an electric field and a magnetic field experiences a force

(9) |

Likewise, a distributed charge distribution of charge density and current density experiences a force density