So let's rewrite this equation again.

D psi of 1,

t is equal to S of 2,

t minus r_1 2 over c times dV2 over 4pi r_1 2.

Adding the contribution from all the pieces of the source

means doing integral over

all regions where S is not equal to zero.

If S is equal to zero then we don't have to do

that because there is no contribution.

So if we do that the psi of 1,

t is equal to integral of S of 2,

t minus r_1 2 over c over 4pi r_1 2 dV2.

That is the field at point one at a time t,

is the sum of all the spherical waves which leaves

a source elements at two at time t minus

r_1 2 minus C. So it's like if you

throw a stone to a calm pond,

you are making spherical waves,

that's one point source.

If you throw a lot of stones,

then those stones will add up,

and this is like this.

The time here is when you threw the stone,

and you are seeing that afterwards.

So this is the solution of

our wave equation for any set of sources,

and we now see how to obtain

a general solution for Maxwell equation.

So for psi is equal to scalar potential phi,

then S is equal to rho over

epsilon nought and for psi is equal to

vector potential a then S is just

equal to j over epsilon nought c squared.

So we can use the same equations

for the two different potentials.

If we know the charge density,

rho of x, y, z,

t and the current density j of x,

y, z, t everywhere,

we can immediately write down the solutions of

the Laplacian of phi minus 1 over c squared

del squared phi over del round T square is equal

to minus rho over

epsilon nought and Laplacian of A minus 1 over c square,

round square A over round t square is equal to

minus j over epsilon nought z squared.

So it is just technicality that matters here.

So if you understand one thing,

you can understand the other thing as well.

So that's the beauty of math,

even though you don't know

the details of what's happening down there,

you have the same governing equations.

If you solve the equations from bottom up, top down,

then you will be able to apply

the knowledge to different fields as

we just discussed before.

You can see those differences in these equations here.

This equation has the c squared and the rho in the j.

Yeah, very good.

So you see rho is here,

instead here is j and c squared.

So with that in mind,

we can solve for both phenomena.

The fields E and B can then be found by

differentiating the potentials that we just solved.

So electric field will be minus del phi minus dA over

dt and B field will be del cross A.

This equation really tells us

that when we think about a ring, a metallic ring,

and if we put a magnet close to

the ring and we shake the magnets

so we are inducing an electromotive force,

then it is really coming from the change of

magnetic potential not the electrostatic potential.

So in a ring,

it is hard to set up

a gradient because the starting point

and ending point is the same.

But you can set up a vector potential time dependence,

so that's how we can understand why

we still have electric field around

the ring that could make a light bulb

lit in this closed loop.

So we have solved Maxwell's equations,

given the currents and charge in any circumstance,

we can find the potentials directly from

these integrals and then

differentiate and get the fields.

So we have finished with the Maxwell theory.

All that remains is to take a moving charge,

calculate potentials from these integrals and then

differentiate to find electric field

from the equation we just mentioned.

The gradient of the electrostatic potential

and the time derivative of vector potential.

Then we will get this equation that we

just mentioned before and discussed in detail.

So Melodie, let me ask you a simple question.

Of these three terms,

what do these two terms mean?

So this first term is the retarded Coulomb field and

then this one corrects it into an instantaneous field,

and together they're like the first of a Taylor series.

Yeah. So as Melodie mentioned,

this is the first-order approximation of Taylor series,

which makes this retarded Coulomb field to

almost instantaneous coulomb field for small distance.

So this will be exactly the same form as we

learned from statics that we have the Coulomb field.

How about the third term?

But still account for

the acceleration and the relativistic effects.

Exactly. So the third term

dominates when we travel long distances

and is more to do with

the light phenomena, wave characteristics.

So you can see it depends on

the acceleration of the charge rather

than the charged quantity

and the distance vector of the charge.

So that's what we have learned,

and this is really the essence of the Maxwell theory.

So we have covered the universe of electromagnetism and

here's the structure built by Maxwell,

complete in all his power and beauty.

So in the previous slides,

we have dealt with

the Maxwell equations in four equations.

Introducing the concept of field;

an electric field and magnetic field and we have also

explained the characters of those fields namely;

the flux and circulation.

In order to understand flux and

circulation of each field,

we came up with divergence and curl.

So divergence of electric field is

here that curl of electric field is there.

Divergence of magnetic field is here

and curl of magnetic field is there.

So with these four equations,

we were able to understand

most of the static and quasi-static phenomena.

Now at this point,

we're at the point to solve

the general equations in the form of this electric field

is the gradient of

the electrostatic potential and

the time derivative of the vector potential,

and magnetic field is curl of magnetic vector potential.

With the Lawrence gauge,

we were able to come up with

this beautiful two sets of equations,

where solving these two will give us

the most rigorous sets of

equations that complete the theory.

However, as you can see

it's still very difficult and complicated.

We will give you some examples that will help

you understand those equations in more details.

So let's think about

the fields of an oscillating dipole as

an example where dipole can move up and down.

By the way Melodie,

what is an electric dipole?

So an electric dipole is when you

have a set of charges so maybe

like positive and negative and there's

a distance between them like here.

So as Melodie draw here,

if you have a set of charges with

equal magnitude and opposite polarity,

the dipole moment is formed and the dipole moment

is defined by the charge times the distance vector here.

So you can see the vector is pointing from minus to plus.

So here we have to limit

ourselves here just to show that in

a few examples this phi (1,t) is equal to integral of

rho (2,t) minus r_12 over C

over 4 pi epsilon nought r_12 times dv2.

A (1,t) is integral j

(2,t) minus r_12 over

c over 4 pi epsilon nought c square,

r_12 times dv2 give the same results as the one that we

just mentioned again and again and

again solved by Richard Feynman.

So first, we will show that that

e equals q over 4 pi epsilon nought

times e_r prime over

r prime square plus r prime over c times d over dt,

e_r prime over r prime square,

plus 1 over c squared d squared over dt

square e_r prime and cB is

equal to e_r prime cross E gives the correct fields for

only restriction that the motion of

the charged particles is non relativistic.

So if the motion of the charged particle is

non-relativistic that means it is moving very slow.

Its motion is very slow.

So we consider a situation

where we have a blob of charge that is moving

about in some way in small region

and we will find the fields far away.

So from a distance,

this blob of charge may look like a point charge,

To put it another way,

we're finding the field at any distance from

a point charge that is shaking up

and down in very small motion.

Since light is usually

emitted from neutral objects such as atoms,

we'll consider that our wiggling charge q is

located near an equal and opposite charges at rest.

So you can imagine you have

protons which is very heavy, fixed,

an electron cloud moving up and down,

that will make your dipole moving up and down.

If the separation between the centers of

the charges d as we mentioned before,

the charge will have a dipole moment

p which is equal to q

times d which we take to be a function of time.