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Maxwell's Equations

• Gauss’s Law allows you to find the electric field in situations of spherical, cylindrical, and planar symmetry.
• Gauss’s Law for Magnetism states that the total magnetic flux through any closed surface is zero, and is a direct outcome to the finding that there are no magnetic monopoles.
• Faraday’s Law allows you to find the inducted emf due to a changing magnetic flux.
• Ampere’s Law allows us to calculate the magnetic field due to an electric current as well as a changing electric field. The contribution due to the penetrating current is known as the conduction current, and the contribution due to the changing electric field is known as the displacement current.

Maxwell's Equations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Objectives 0:07
• Maxwell's Equations 0:19
• Gauss's Law
• Gauss's Law for Magnetism
• Ampere's Law
• Revising Ampere's Law 1:49
• Allows Us to Calculate the Magnetic Field Due to an Electric Current
• Changing Electric Field Produces a Magnetic Field
• Conduction Current
• Displacement Current
• Maxwell's Equations (Complete) 2:58

Transcription: Maxwell's Equations

Hello, everyone, and welcome back to www.educator.com.0000

In this lesson, we are just going to review Maxwell’s equations.0004

Our objectives include making sure students are familiar with these equation so they can associate each equation with its implications.0008

Let us take a look at Maxwell’s equations as we define them so far.0015

First we have Gauss’s law, the integral / the closed surface E ⋅ DA is the total close charge divided by ε₀.0021

This was very useful, although it is always true, it is mostly useful0030

when you are looking for the electric field indications where you have some sort of symmetry.0034

Symmetry typically being planar, cylindrical, or spherical.0040

Gauss’s law for magnetism, the integral / the close surface of B ⋅ DA = 0.0044

It was another way of stating that magnetic monopoles do not exist.0050

Any closed surface whatever magnetic field lines go in, the same amount of magnetic field lines come out.0054

Faraday's law, the integral / the closed loop of E ⋅ DL = - D/ DT the derivative/ the open surface of B ⋅ DA,0061

this is really the magnetic flux.0070

When we have no change in magnetic flux, this simplified down to Kirchhoff’s voltage law.0072

Ampere’s law, we said the integral / the closed loop of B ⋅ DL = μ₀ I.0079

We use this to find the magnetic field in situations of symmetry that was a much more elegant than0085

when we used the more challenging Biot-Savart law.0092

We put the asterisk here, there was more that we had to talk about it.0096

By the way, that is penetrating current.0100

There is a little bit more to ampere’s law and that is what we are going to develop next.0104

Ampere’s law as written, allows us to calculate the magnetic field due to some electric current that penetrates our Amperian loop.0109

However, we also know that the change in electric field produces a magnetic field and we have not taken that into account yet.0117

That piece looks like this, the integral/ the closed loop of B ⋅ DL is the permeability × the permittivity × the time rate of change of the electric flux.0124

Or if you wanted to expand out our electric flux, μ₀ ε₀ × the derivative of and0136

there is our electric flux, integral / the open surface of E ⋅ DA.0141

That is the piece that we have not thrown into the equation yet, even though we know it exists.0147

How do we put all that together?0152

To combine the effects, we are going to take a look at the contribution due to the penetrating current0155

which we are going to call conduction current.0161

The contribution due to the changing electric field, the changing electric flux, and that is what we called the displacement current.0164

Putting all of those together to come up with a refined version of Ampere’s law, our final Maxwell's equation looks like this.0172

Gauss’s law for magnetism, faraday’s law, and Ampere’s law, now the integral / the closed loop of B ⋅ DL = μ₀ I penetrating,0183

our conduction current piece + our displacement current piece μ₀ ε₀ × the derivative of the electric flux.0194

That is our complete Maxwell’s equations and that completes the basic content of the course for E & M AP Physics C.0205

Thank you so much for watching www.educator.com.0214

Make it a great day everybody.0217