So let's take a look at the comparison between neutron diffusion and electrostatics.

So if there are sources in the volume that

generate S neutrons per unit time in a unit volume,

then the net flow out of Delta V will be described like this,

and again if it is a steady state,

we will remove this term which is

related to the change of the neutron density in a given volume.

If you do that, you will see that the divergence of D Del N is equal to minus S,

and if you compare this with the electrostatic equation,

you can replace one by one.

So in this case, Melody.

What does N correspond to, what is it?

Also corresponds to the potential.

Exactly. So, if we solve the equation for potential of charged sphere,

then you can imagine or you can already know,

now we can also get the neutron density as a function of the distance in this case.

So indeed, these are two schematics as you can see,

this is the uniform sphere of charge,

and describing the potential,

you will see you have quadratic function here and then you have 1 over r function.

For the neutron case,

we will guess it will have the same shape of curve.

So we will solve what is the density of neutrons everywhere,

what is the ratio of neutron density at the center to

the nutrient density at the surface of the source region using this analogy.

You may say, if you are creating neutrons at the same rate everywhere,

the density of neutron should be uniform.

That might be a naive guess,

but in fact, you can already see here that's not the case.

So here, we're going to first

revisit the problem of electric potential inside

and outside of a uniformly charged sphere,

as you can see here,

where density is the same and see how the potential varies as a function of distance.

Before solving this, let me ask my teaching assistant,

Melody, what type of strategies or methods we can use to tackle this problem?

Okay. So, we're looking at essentially two different regions here,

one is within this sphere and one is outside of the sphere.

So within this sphere,

it depends on radius,

because as you expand the radius,

the amount of charge inside is increasing.

However, outside of the sphere,

there are no additional charges added.

So, it only depends on what's inside the sphere.

Exactly. So like Melody said,

we will rely on the fact that outside the sphere,

if we make a Gaussian surface larger than

the uniform sphere of charge or uniformly charged sphere,

then we can think this as a point charg.,

then the relationship will be 1 over r, the potential,

but if we use Gaussian surface smaller than this sphere,

then the charge that the Gaussian surface

can contain depends on the radius of the Gaussian surface.

So we will have totally different r dependence.

So using those conditions and method of Gauss law,

we are able to find the potential outside will depend on 1 over r like a point charge,

and as you can see,

you will be able to get Rho a cubic over three Epsilon not, as a coefficient.

For points inside as mentioned before,

the electric field will be linearly increasing as

a function of radius because of the reasons Melody just mentioned.

So then we can integrate it to get a quadratic form of r dependence with a constant,

and at the interface,

at the surface because these two should have the same value,

we can get the unknown constant,

C, using that boundary conditions.

When we do that, we find that the C is equal to Rho a squared over two Epsilon not.

So if we put that back into our equation,

the potential inside will have a quadratic function that is described here.

So you can see you have

a very smooth continuous function where the first part have a quadratic dependence,

and then the second part have 1 over r dependence.

So, neutral density inside and outside a source can have

the same solution because the equation that

governs the phenomenon it has the same form as electrostatics,

so you just replace one by one.

For example, the potential by nutrient density,

and then you can get the same type of formula

and then plot the density as a function of distance.

As you can see, at the center,

you have the largest neutron density.

As you go to the surface,

it decreases as a function of r. So,

a uniform source doesn't produce a uniform density of neutrons.

That's what we learned from this equation, and also,

we can know the ratio of N at the center to that at

the edge which is 3 over 2 which is 1.5.

So you have 50 percent more nutrients in the center when compared with the surface.

All right, let's take another example.

So in this case is it's a rotational fluid flow,

the flow past a sphere.

So imagine in your bathtub,

you somehow dropped a bead a glass bead,

and then you wonder what would be the velocity vector around this bead.

If that's really something you want to solve,

then you can solve it using the Poisson equation that we learn with some assumptions.

So let's consider an example of dry water which is an incompressible.

Non-viscous and circulation free liquid,

and we represent the flow by giving the velocity vector v of

r as a function of position r and if the motion is steady,

v is independent of time.

So you can see the divergence of

density times the velocity vector is equal to minus round rho over round t,

but because the density doesn't change as a function of time, it should be zero.

So in this case, you can see the divergence of velocity will be zero,

and if there is no circulation of velocity like in

the drain where you have a whirl-type of vector,

then it will be zero.

So then you can see this is really like electrostatics,

and we already learned if a vector field has circulation-free condition,

we can make a relationship with a potential.

So, we can have a relationship of v equals minus del phi and therefore if we put it back,

it becomes like a Poisson's equation with net charge density zero.

So, with that in mind,

let's take a look at this picture again.

Imagine, instead of ball moving,

the water is moving.

Everything is relative.

So, in this way,

we can imagine this in a better simpler term.

Now, as you look at this field line, what do you think?

Can you think of a situation in electrostatics that is similar to that,

say you have uniform electric field and you are putting some sphere inside,

and do you think you can create this kind of situation?

Let's ask my teaching assistant Melody.

So, Melody, can we make this kind of situation

using say metal sphere or dielectric sphere?

I think maybe this would be the case where the dielectric constant is zero again.

I'm not sure.

Yeah, and that's a very good guess.

So dielectric constant zero doesn't exist.

So, then you may ask,

what if I have metal sphere? What happens?

How does the electric field change?

So let me ask Melody.

If I put metal sphere inside uniform electric field, what happens?

Yeah. So, if you fit the metal sphere inside there,

then the charges will accumulate,

and they would create an electric field within the sphere that cancels out.

So, if I have a metal sphere,

as Melody mentioned, you cannot have electric field inside the metal sphere.

So, the field that want to penetrate will be stopped here.

It will not be able to penetrate further because

you are creating a situation where you have a plus charge accumulated,

minus charge accumulated, so you annihilate.

Then you have penetrating electric field from here.

Now, if I have a field from a sideway,

and if I create a lateral field here, what happens?

It doesn't feel happy.

So, it will make sure any field that borders will only come in normal direction,

and the same goes here and then same goes there and same goes there.

So you will see the shape of this field

is only saying that you don't have any field inside the ball that's the same.

But outside you have a different boundary conditions where in this case,

you only have normal component,.

In that case, you only have in-plane component.

So, how can we understand this?

So, as Melody already mentioned,

this can only be solved with unusual conditions,

where you have almost zero dielectric constant or negative dipole moment,

which we will discuss shortly.