Melody, what will happen to polarization?

Polarization will increase.

Polarization will increase.

So, let's think about this,

if I apply more or stronger electric field they will tend to align more.

So, the polarization will increase, that's understandable.

Now, if I increase temperature,

what happens to my polarization?

It decreases.

It decreases. So, if I heat the system up,

they want to randomize more,

they don't want to align with the electric field.

So, I can make a joke out of this.

So, if somebody, the leaders want to align

the values of the society applying big pressure or electric field they can do it,

but if the society has high temperature where they have more freedom of expression,

where they tend to think more freely,

then it's very hard to align.

If you lower the temperature,

it's easier to align.

Okay. So, here's we're going to discuss Langevin Function,

and from this discussion,

I hope you are able to solve the homework problem that we just discussed.

Langevin function is a function for magnetic moments,

but as you can see,

magnetic moments behave similarly to electric dipole moment.

So, if you just replace the parameters in

this equation you can solve the problem for electric dipole moment as well.

So, let's take a look at this.

For paramagnetic substance with

magnetic moments M in an applied field H at temperature T,

then if we make the assumption that there is no preferred alignment with the substance,

we can assume that the number of moments N of theta again,

the same between angles theta and theta plus D theta with respect to

the external magnetic field H is proportional the solid angle two pi sine theta d theta.

Then the probability density function where you can see

the net magnetic moment over the maximum moment is equal to this function.

Where you see the denominator is a whole number of dipole moments,

and the nominator is the dipole moment that is within the strip of the same solid angle.

If you do this calculation using Boltzmann equation as we did,

where the distribution function follows the Boltzmann function.

Then without doing the approximation,

you can do the mathematics for exponential function by

doing a partial integration and then you will get this function which will

result in hyperbolic cotangent alpha minus one over

A where you see the A function is MU naught H over KT,

and MU naught H is something similar to

external field times the P_naught, the dipole moment.

So, if you replace those parameters with the parameters we discussed,

then you will have the same function.

The graphical representation of Langevin function is like this.

As you can see, in the electric field case,

if you increase the electric field,

you have linearly increasing portion and then it goes way off from

this linear portion and saturates to the asymptotic values here.

So, the portion that we just solve together was this starting point.

So, that's the same thing,

if I apply huge magnetic field to magnetic dipole moment they tend to

align but if the temperature increases they tend to be randomized.

Okay. So, as you can see from this graph,

the slope of the initial part of the graph is

exactly the same as the one that we derived,

the polarization is equal to N P_naught square times E over three KT.

So, you can see the polarization is proportional to the field E. So,

there will be normal dielectric behavior,

we also expect that p depends inversely on T because at higher temperature,

there is more disalignment by collisions.

So that's, one over T dependence and which is called Curie's law.

So, Curie's law is permittivity,

is the inverse permittivity or susceptibility is proportional to one over T. Okay.

Let me ask Melody, what susceptibility was?

The susceptibility was the influence

that a particular system would get from an electric field.

Exactly. So, let me explain what Melody just explained to us in equations.

So, P polarization is equal to

susceptibility times permittivity of vacuum times electric field.

If you compare this equation with this equation,

what you see is the susceptibility is within this side.

In other words, susceptibility is proportional to one over T,

and that is exactly what happens for paraelectric materials,

it falls off exponentially as a function of temperature from the Curie temperature,

meaning at Curie temperature susceptibility will diverge to infinity.

So, from this simple Langevin Function we can get many messages

or knowledge from this simple equation.

Now, why do we have P_naught square dependence here?

Why do we have P_naught square?

So, let's think about it.

In a given electric field,

the aligning force depends on P_naught we know that.

Right? The mean moment that is

produced by the lining up is again proportional to P_naught,

so that's why we have P_naught square.

So, this is a very important factor in determining the polarization of the system.