# Maxwell Equations Revision Notes

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16.18Maxwell Equations

In these revision notes for Maxwell Equations, we cover the following key points:

• What does Gauss Law for electric field say?
• The same for magnetic field?
• How can we induce a magnetic field inside the plates of a charged capacitor?
• What does Ampere-Maxwell Law say?
• What are the special cases of Ampere-Maxwell Law?
• How does the magnetic field inside the capacitor change with the distance from the centre of plates?
• What is the displacement current? How is it related to the real current used to charge the capacitor?
• How to Find the Induced Magnetic Field?
• What are the Maxwell Equations? What do we find using them?
• Why Maxwell Equations are so important?

## Maxwell Equations Revision Notes

The Gauss Law for the magnetic field implies that magnetic monopoles do not exist. Mathematically, this law means that the net magnetic flux Φm through any closed Gaussian surface is zero. The Gauss Law formula for magnetic field is

ΦM = B ∙ dA = 0

The Gauss law for electric fields on the other hand, is

Φe = E ∙ dA = Q/ε0

A changing magnetic flux induces an electric field in a loop, which on the other hand generates an emf (and current) in the loop. This represents the Faraday's Law of Induction the integral form of which, is

E dL = -b/dt

The reverse is also true. A changing electric flux can induced a magnetic field. The equation representing this property is known as Maxwell Law of Induction and the corresponding equation is

B dL = μ0 ∙ ε0e/dt

Here B represents the magnetic field induced along a closed loop by means of the changing electric flux ΦE in the region enclosed by the loop.

The mathematical form of Ampere-Maxwell Law is

B dL = μ0 ∙ ε0e/dt + μ0 ∙ iencl

As special cases of this law, we can consider the two following scenarios:

When there is a current but no change in electric flux (such as a wire carrying a constant current), the first term on the right side of the Ampere-Maxwell Law is zero. As a result, the Ampere's Maxwell Law reduces to Ampere's law of Induction we have seen in tutorial 16.6.

B dL = μ0 ∙ iencl

When there is a change in electric flux but no current (such as inside or outside the gap of a charging capacitor), the second term on the right side of the Ampere-Maxwell Law becomes zero, and so the Ampere's Maxwell Law reduces to the Maxwell's Law of induction

B dL = μ0 ∙ ε0e/dt

The magnetic field outside the plates of a charging capacitor decreases when the distance from the centre of plates increases. However, even between the plates the magnetic field is very small when compared to the magnitude of electric field. This is the reason why capacitors are considered as electric devices and not as magnetic ones.

When analysing the equation derived from the Ampere-Maxwell Law

B dL = μ0 ∙ ε0e/dt + μ0 ∙ iencl

it is obvious that the expression

ε0e/dt

must have the dimension of current. This current is known as the "displacement current, id", despite no current is really being displaced. Thus, the above equation becomes

B dL = μ0 ∙ id,encl + μ0 ∙ iencl

The real current I which charges the capacitor is

dQ(t)/dt = I = ε0 ∙ A ∙ dE(t)/dt

As for the displacement current Id, we have

id (t) = ε0e/dt
= ε0d(E ∙ A)/dt
= ε0 ∙ A ∙ dE(t)/dt

Thus, since the expressions for the true and fictitious currents are identical, we can consider the fictitious displacement current Id simply as a continuation of the real current I from one plate of capacitor to the other plate flowing through the gap between plates. Although no charge actually moves across the gap between the plates, the idea of the fictitious current Id can help us to quickly find the direction and magnitude of an induced magnetic field.

The magnetic field produced by the displacement current inside the capacitor plates, at a distance r from the central axis (r < R), is

B = μ0 ∙ id/2π ∙ R2 ∙ r

and the magnetic field outside the plates (at r > R from the central axis), is

B = μ0 ∙ id/2π ∙ r

The four Maxwell Equations are:

### Gauss' Law for electricity:

E ∙ dA = Qencl/ε0

It shows the relationship between net electric flux and net enclosed electric charge.

### Gauss' Law for magnetism:

B ∙ dA = 0

It shows the relationship between net magnetic flux and net enclosed magnetic charge.

E dL = -b/dt

It shows the relationship between induced electric field and changing magnetic flux.

### Ampere-Maxwell Law

B dL = μ0 ∙ ε0e/dt + μ0 ∙ iencl

It shows the relationship between induced magnetic field and changing electric flux and to current.

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