Maxwell's Equations

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1st Maxwell equation

Flux is just field * m

Law

(see integral)

Intuition

This means E-field is nothing but electric flux density. From Interpretation of Integrals, we can easily see how Gauss's law holds.

Remark

"True" from of Gauss's law is:


where .

Intuition

The units of end up being coulombs per meter squared. In fact, is nothing but charge density (like ). Once again, from Interpretation of Integrals, we can see how Gauss's law is formed.

Gauss's Law for Magnetism (Second Maxwell Equation)

See Gauss's law

Magnets have a north and south pole. If you break a magnet in half, each piece has its own north and south pole. In the classical world, magnets cannot be monopole. This means the magnetic flux on a closed surface must be 0, and we have our second Maxwell equation:

Equation

(see integral)

Pre: Current Density

Consider a wire which has been bent into a loop. Then:

Law

(see integral, B-field, dot product)

We call the loop an Amperian loop, and it acts as a boundary, similar to a Gaussian surface. For Ampere's law, the shape of the loop does not matter, as long as it is closed.

Similar to Gauss's law, if we can create a mathematical loop over which we can guarantee one of the following:

  1. is in the direction of throughout the loop and is a constant through out the loop
  2. is 0 in parts where is constant (i.e OR )

then we can use Ampere's law to calculate the magnetic field for the current distribution.

With these conditions met, we have:

Faraday's Law

Law

Consider a closed loop, and the open surface made by this loop.

Faraday's law states that if the magnetic flux enclosed within the loop changes with time, then an E-field is created which loops around the flux.
Furthermore, the strength of this E-field is proportional to the rate of change of the enclosed magnetic flux.

The negative sign is due to Lenz's Law.

(see integral, dot product, derivative)

Note the connection to Ampere's law, which states that . If we walk around loop and integrate , we get the total current enclosed by .

Faraday's law has the same form as Ampere's law, except with a time derivative. A time-varying magnetic flux creates an electric field which curls around the magnetic field.

Faraday's Law is Non-Conservative

For electrostatic fields, we have:

If we start from a point, and walk on a closed loop and come back to the same point, we do not create a potential difference.

However, if the magnetic field is time-varying and we walk on the loop and then return to the same point, we create a potential difference across the same point.

Important

Faraday's law is non-conservative. That is, path matters in Faraday's law.

Remembering

We have the following combinations:

We have every possible combination. Now we just have to define each law.

Note the last two laws each involve non-closed surface integrals, which is important. Just remember that B and E are flux density and J is current density, and with Interpretation of Integrals it's not hard to remember.

Other strategies:

  1. Ampere's law involves Ampereian loop, hence
  2. Potential difference is also equal to , like Faraday's law
  3. In the same way Gauss's law states , we also have (because E and B are flux density by definition)