Hello everybody.

Welcome to Electrodynamics

and Its Applications,

this will be the 17th lecture.

My name is Professor Seungbum Hong,

and to my right side, I have

my teaching assistant Melodie Glasser.

So, today, we will

cover the complete set

of Maxwell's equations.

Up to now, we have come here inch by inch,

having a lot of constraints and assumptions,

but now we will come back to

the complete set for Maxwell equations.

We have the complete and correct story for

electromagnetic fields that may be

changing with time in any way.

So, as you can see in this table below,

the complete equations are written here.

So, Melodie, this was

a long journey, right?

Right.

We're almost to the end of

our full specialization lecture stairs,

and this will be marked as

a big victory for both of

us and for our students

who are listening to this lecture.

So, the first equation, if you remember,

is the divergence of

electric field is equal to

the charge density divided by

the permittivity in vacuum.

In other words, you can

interpret that as the flux

of the electric field

through any closed surface

is equal to the charge

enclosed by the surface

divided by the permittivity.

The second equation would

be the curl of electric field

is equal to the time derivative

of magnetic field,

and we put a minus sign in front of it.

So, the meaning would

be the line integral

of electric field through

a closed surface is equal to

the time change of

flux of B field through the loop.

The third is there is

no magnetic monopole that

is written here like

divergence of B field is always zero,

so the flux of B through

any closed surface should always be zero.

So, if there's any in-flux,

there should be the same

and equal amount of out-flux.

The fourth equation is

one of the equation that Maxwell

really had a significant contribution

by adding a new term

to what was known that time,

and this is the curl of

magnetic field is equal to two terms.

One is the current through the loop,

current density at the point of

interest divided by the permittivity

in vacuum plus the change

of electric field at the point of interest,

and that could be interpreted as

integral B on around a loop is equal

to the current through the loop divided

by Epsilon naught,

plus the time change

of flux of electric field through the loop.

Okay. So, let me

ask my teaching assistant among those four,

what is your most favorite equation.

I like Gauss's Law,

which is the first one.

Here is the first one we looked at,

and so that looks okay.

May I ask the reason.

I just like it because we've used it

the longest and I

think maybe it's less complicated to

look at in doing calculations.

Exactly. In fact many of

semiconductor engineer or device designers

use this first law which will

be changed into

the Poisson's equations in one dimension,

and then they will

work on this charge density,

which is not only coming from the electrons

in the metal or holes in the metal,

but also ions or defects that are included,

then this equation also

becomes a little bit complicated.

Okay. So, now we're at the point to discuss

the equations for classical physics.

So, we know there

are three conservations law,

one of which is conservation of charge,

and that could be nicely written by

this differential equation

in vector calculus,

where you see on the left side,

the divergence of current density is

equal to the minus

of the change of

the charge density as a function of time,

and this could be written

as the flux of j

through a closed surface is

equal to the amount

of charge that is inside

that surface which change as

a function of time

and that time derivative,

the negative times

derivative of this charge,

will be equal to that.

So, this would be a very important law.

We also learned about

force law, where we know,

if you know the position

and velocity of the charge,

and the electric field and

magnetic field at the point of interest,

then we can know

the force imposed on the charge,

and the law of motion states

that that force should be

equal to the change of

momentum as a function of time,

that will be the time derivative of

the momentum where the momentum

is defined by mv over

square root one minus v

squared over c squared,

which is the Newton's law with

Einstein's modification for the mass,

and also we know the law

of gravitation which resembles

the Coulomb's law in electrostatics.

So, let us focus

on Maxwell's equations more,

and we will discuss first with

the Melodie's favorite equation number

one and equation number three.

So, let's take a look

at equation number one again,

which is the Delta E is

equal to rho divided by Epsilon naught,

and this first equation

shows that the divergence of

E is the charge density

over Epsilon naught,

and this equation is always true,

not only true for statics but

also dynamics and true in general.

In both dynamic and static fields,

Gauss' law is always valid.

So, let me ask

my teaching assistant Melodie,

again, what Gauss' Law was.