# A question about the fields D and H for fields in matter

• Coffee_
In summary: This formula seems to only care about what the macro current is through the loop. For example, I take a toroid-shaped solenoid and put material inside this solenoid and then take the loop integral of H. I will find that the field H does not depend on the material inside the solenoid and only on the properties of the solenoid itself. The previous poster seemed to imply this was the opposite case for H fields.No, it is not the opposite case. The two cases are both valid. No, it is not the opposite case. The two cases are both valid.

#### Coffee_

Let me give you an example of a reasoning I made in a simple case. Afterwards comes the question:

Start of reasoning.

Consider two standard ideal capacitor plates with a dielectric material in between them. Let's call the external field caused by the plates ##\vec{E_{ext}}## and the average macroscopic internal field caused by polarisation ##\vec{E_{int}}## The total field at any point is then on average ##\vec{E_{tot}}=\vec{E_{ext}}+\vec{E_{int}}##

One can prove that for homogenous polarisation ##\vec{E_{int}}=-\frac{\vec{P}}{\epsilon_{0}}## and thus ##\vec{E_{tot}}=\vec{E_{ext}}-\frac{\vec{P}}{\epsilon_{0}}##

Let's now look at the ##\vec{D}## field at any point in the material. The definition is ##\vec{D}=\epsilon_{0}\vec{E_{tot}}+\vec{P}##. Plugging the previous expression into D, results in:

##\vec{D}=\epsilon_{0} \vec{E_{ext}}##

End of reasoning.

QUESTION:

1) So in this case ##\vec{D}## seems to be not only independent of polarization but really equal to the external field up to a constant. Is this the correct way to interpret the ##\vec{D}##-field, as something that is equal to the E field that is caused by sources of free charges and neglect any E fields caused by polarization?

2) If I replace D with H and E with B , is the same interpretation correct?

Coffee_ said:
1) So in this case ##\vec{D}## seems to be not only independent of polarization but really equal to the external field up to a constant. Is this the correct way to interpret the ##\vec{D}##-field, as something that is equal to the E field that is caused by sources of free charges and neglect any E fields caused by polarization?
The polarization changes E, and D is defined specifically to be invariant of that (as long as the setup is as nice as yours).
2) If I replace D with H and E with B , is the same interpretation correct?
D with B and E with H. Yes, that is not intuitive.

Edit: removed weird sentence fragment

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Coffee_ said:
mfb said:
For fields orth. The polarization changes E, and D is defined specifically to be invariant of that (as long as the setup is as nice as yours).
D with B and E with H. Yes, that is not intuitive.

May I ask what you mean by your last line? Is H no longer invariant in nice setups?

B takes the place of D, and H the one of E, so yes.

In general, the "source" of D is not just the free charges.

The polarization P will also have an effect on the what the value of D will be. For certain simple cases with lots of symmetry, D does end up being the same as if you treated only the free charges as "producing" D. But, in general, this is not true. Examples where D depends on the properties of the dielectric as well as the free charges can be found in standard textbooks.

It is true that the divergence of D is given by ##\vec{\nabla} \cdot \vec{D} = \rho_{free}##, but that equation is not sufficient to find D except for simple configurations with sufficient symmetry. In general, a vector field is determined by both its divergence and curl. You can see that the curl of D depends on the curl of P. So, P can act as a "source" for the curl of D.

TSny said:
In general, the "source" of D is not just the free charges.

The polarization P will also have an effect on the what the value of D will be. For certain simple cases with lots of symmetry, D does end up being the same as if you treated only the free charges as "producing" D. But, in general, this is not true. Examples where D depends on the properties of the dielectric as well as the free charges can be found in standard textbooks.

It is true that the divergence of D is given by ##\vec{\nabla} \cdot \vec{D} = \rho_{free}##, but that equation is not sufficient to find D except for simple configurations with sufficient symmetry. In general, a vector field is determined by both its divergence and curl. You can see that the curl of D depends on the curl of P. So, P can act as a "source" for the curl of D.

Thanks. Same reasoning applies to B and H right?

Coffee_ said:
Thanks. Same reasoning applies to B and H right?
Yes, that's right.

TSny said:
Yes, that's right.

May I then ask about the formula (assuming no E field is changing) ''closed loop integral of H'' = sum of macroscopic currents through the loop.

This formula seems to only care about what the macro current is through the loop. For example, I take a toroid-shaped solenoid and put material inside this solenoid and then take the loop integral of H. I will find that the field H does not depend on the material inside the solenoid and only on the properties of the solenoid itself. The previous poster seemed to imply this was the opposite case for H fields. Am I misunderstanding something?

You have a different setup now, a solenoid is not a capacitor.
H does not change in that case, but B does.

## 1. What is the difference between the fields D and H for fields in matter?

The fields D and H are two separate components of the electromagnetic field in matter. D represents the electric displacement field, which is the amount of electric charge in a given volume of material. H represents the magnetic field intensity, which is a measure of the magnetic force exerted on a unit magnetic pole in a material. In simple terms, D is related to the behavior of electric charges in a material, while H is related to the behavior of magnetic poles.

## 2. How are the fields D and H related to each other?

The relationship between D and H is described by the Maxwell's equations, which state that the electric displacement field (D) is equal to the sum of the electric field (E) and the polarization of the material (P). Similarly, the magnetic field intensity (H) is equal to the sum of the magnetic field (B) and the magnetization of the material (M). Therefore, D and H are directly related to the properties of the material they are in.

## 3. Are the fields D and H affected by the type of material they are in?

Yes, the fields D and H are affected by the type of material they are in. This is because different materials have different properties, such as their dielectric constant and magnetic permeability, which determine how the fields will behave in that material. Therefore, the values of D and H can vary depending on the type of material they are in.

## 4. Can the fields D and H be measured?

Yes, both D and H can be measured using appropriate instruments. D can be measured using a device called an electric field meter, while H can be measured using a magnetometer. These measurements are important in understanding the behavior of the fields in different materials and in the study of electromagnetism.

## 5. How do the fields D and H affect the behavior of light?

The fields D and H play a crucial role in the propagation of light in a material. They determine the speed at which light travels through a material, as well as its direction and polarization. The interaction between D and H also affects how light is reflected, refracted, and absorbed by a material. This is why the fields D and H are important in the study of optics and the behavior of light in different materials.