The first equation is simply Gauss' law (see Sect. 4). This equation describes how electric charges generate electric fields. Gauss' law states that:

This can be written mathematically asThe electric flux through any closed surface is equal to the total charge enclosed by the surface, divided by .

(305) |

where is a volume bounded by the surface , and is the

The second equation is the magnetic equivalent of Gauss' law (see Sect. 8.10). This equation describes how the non-existence of magnetic monopoles causes magnetic field-lines to form closed loops. Gauss' law for magnetic fields states that:

This can be written mathematically asThe magnetic flux through any closed surface is equal to zero.

where is a closed surface.

The third equation is Faraday's law (see Sect. 9.3). This equation describes how changing magnetic fields generate electric fields. Faraday's law states that:

This can be written mathematically asThe line integral of the electric field around any closed loop is equal to minus the time rate of change of the magnetic flux through the loop.

where is a surface attached to the loop .

The fourth, and final, equation is Ampère's circuital law (see Sect. 8.7). This equation describes how electric currents generates magnetic fields. Ampère's circuital law states that:

This can be written mathematically asThe line integral of the magnetic field around any closed loop is equal to times the algebraic sum of the currents which pass through the loop.

(309) |

where is a surface attached to the loop , and is the

When Maxwell first wrote Eqs. (306), (307), (308),
and (310) he was basically
trying to summarize everything
which was known at the time about electric and magnetic fields
in mathematical form. However, the more Maxwell looked at his
equations, the more convinced he became that they were incomplete.
Eventually, he proposed adding a new term, called the *displacement
current*, to the right-hand side of his fourth equation. In fact, Maxwell was able
to show that (306), (307), (308),
and (310) are *mathematically inconsistent* unless
the displacement current term is added to Eq. (310). Unfortunately,
Maxwell's
demonstration of this fact requires some advanced mathematical
techniques which lie well beyond the scope of this course.
In the following,
we shall give a highly simplified version of his derivation of the
missing term.

Consider a circuit consisting of a parallel plate capacitor of capacitance in series with a resistance and an steady emf , as shown in Fig. 52. Let be the area of the capacitor plates, and let be their separation. Suppose that the switch is closed at . The current flowing around the circuit starts from an initial value of , and gradually decays to zero on the RC time of the circuit (see Sect. 10.5). Simultaneously, the charge on the positive plates of the capacitor starts from zero, and gradually increases to a final value of . As the charge varies, so does the potential difference between the capacitor plates, since .

The electric field in the region between the plates is approximately uniform,
directed perpendicular to the plates (running from the positively charged plate
to the negatively charged plate), and is of magnitude . It follows
that

(311) |

(312) |

(313) |

Equation (314) relates the instantaneous current flowing around the
circuit to the time rate of change of the electric field between the
capacitor plates. According to Eq. (310), the current flowing around the
circuit generates a magnetic field. This field circulates around the
current carrying wires
connecting the various components of the circuit. However, since
there is no actual current flowing between the
plates of the capacitor, no magnetic field is generated in this region,
according to Eq. (310).
Maxwell demonstrated that for reasons of mathematical self-consistency there
must, in fact, be a magnetic field generated in the
region between the plates of the capacitor.
Furthermore, this magnetic field must be the same as that which
would be generated if the
current (*i.e.*, the same current as that which
flows around the rest of the circuit)
flowed between the plates. Of course, there is no actual current flowing between
the plates. However, there is a changing electric field. Maxwell argued that a
changing electric field constitutes an effective current (*i.e.*, it
generates a magnetic field in just the same manner as an actual current).
For historical reasons (which do not particularly interest us at the moment), Maxwell
called this type of current a *displacement current*.
Since the displacement
current flowing between the plates of the capacitor must equal the
current flowing around the rest of the circuit, it follows from Eq. (314)
that

Equation (315) was derived for the special case of the changing electric
field generated in the region
between the plates of a charging parallel plate capacitor. Nevertheless, this equation turns out to be completely general. Note that
is equal to the electric flux between the plates of
the capacitor. Thus, the most general expression for the displacement current
passing through some closed loop is

(316) |

According to Maxwell's argument, a displacement current is just as effective at generating a
magnetic field as a real current. Thus, we need to modify Ampère's
circuital law to take displacement currents into account. The modified
law, which is known as the *Ampère-Maxwell law*, is written

This can be written mathematically asThe line integral of the electric field around any closed loop is equal to times the algebraic sum of the actual currents and which pass through the loop plus times the displacement current passing through the loop.

(317) |

where is a surface attached to the loop .

Equations (306), (307), (308), and (318) are known collectively as
*Maxwell's equations*. They constitute a complete and mathematically
self-consistent description of the behaviour of electric and magnetic
fields.