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Why do people ignore polerization and magnetization when rewriting maxwells equations for different fields?
Thanks!
Thanks!
I'm sorry, I don't understand what you're referring to here. Can you give an explicit example?Isaac0427 said:rewriting maxwells equations for different fields?
Note: sorry, I'm having trouble with LaTeX today. For convenience, I am also going to have the electric constant be just ε.jtbell said:I'm sorry, I don't understand what you're referring to here. Can you give an explicit example?
But what does this have to do with my question?blue_leaf77 said:The auxiliary equation reads ##\mathbf D = \epsilon \mathbf E = \epsilon_0 \mathbf E + \epsilon_0 \chi \mathbf E= \epsilon_0 \mathbf E + \mathbf P## where ##\epsilon = (1+\chi)\epsilon_0## and ##\mathbf P = \epsilon_0 \chi \mathbf E##.
Why don't you read my entire second post before you respond to it. Maybe the top of the post.blue_leaf77 said:Your last equation in post #3 is wrong, in the RHS it should be ##\epsilon_0## instead of ##\epsilon##.
? So what, I don't see any possible relation between that sentence with your doubt about the absence of P in Maxwell equations.Isaac0427 said:I am also going to have the electric constant be just ε.
It's unnecessary to write Gauss equation in terms of P unless you are asked to find this quantity. If you want to express it in terms of P, you would end up with the third equation but with ##\epsilon## replaced by ##\epsilon_0##.Isaac0427 said:Again, can people just please answer my question- where are the P and M terms in some of maxwells equations, such as Gauss' law with the E field?
So the third equation in post number 3 is correct (keeping in mind that I expressed the electric constant without subscripts)? Why is it unnecessary? Wouldn't that be ignoring a term in the equation? Since when is it ok to leave out a term and call the equation correct?blue_leaf77 said:It's unnecessary to write Gauss equation in terms of P unless you are asked to find this quantity. If you want to express it in terms of P, you would end up with the third equation but with ϵϵ\epsilon replaced by ϵ0ϵ0\epsilon_0.
You have to include the subscript in the last equation to make it different from the electric constant in the second equation. The epsilon in the second equation is the one inside a matter while that in the last equation is for vacuum.Isaac0427 said:So the third equation in post number 3 is correct (keeping in mind that I expressed the electric constant without subscripts)?
Since the dooms day later may be?? No terms are being left out, instead they are only hidden in the appearing term.Isaac0427 said:Since when is it ok to leave out a term and call the equation correct?
The Wikipedia article on maxwells equations says that the epsilon in my second equation is also with a subscript 0.blue_leaf77 said:You have to include the subscript in the last equation to make it different from the electric constant in the second equation. The epsilon in the second equation is the one inside a matter while that in the last equation is for vacuum.
Since the dooms day later may be?? No terms are being left out, instead they are only hidden in the appearing term.
I have got the impression that you haven't seen any equivalence between these two relations:
$$\int\int \mathbf E \cdot d\mathbf S = \frac{1}{\epsilon} \oint \rho_f dV$$
and
$$ \int\int \mathbf E \cdot d\mathbf S = \frac{1}{\epsilon_0} \left( \oint \rho_f dV - \int\int \mathbf P \cdot d\mathbf S \right)$$
with ##\epsilon = (1+\chi)\epsilon_0##, see these epsilons are different.
Look at eq. 23 in http://www-users.aston.ac.uk/~pearcecg/Teaching/PDF/LEC2.PDF .
Thank you, this clears everything up!axmls said:You're dealing with different charge densities in each equation. When you write $$\oint \vec{E} \cdot d\vec{A} = \frac{1}{\epsilon_0} \iiint_\Omega \rho \ dV$$ the ##\rho## is the total charge, i.e. the free charge plus the bound charge, whereas the proper equation using the ##D## field is $$\oint \vec{D} \cdot d\vec{A} = \iiint_\Omega \rho_f \ dV$$ using only the free charge.
More explicitly, starting from the equation for the ##D## field, let ##\vec{D} = \epsilon_0 \vec{E} + \vec{P}##, so substituting this in:
$$\oint \vec{D} \cdot d\vec{A} = \iiint \rho_f \ dV \Rightarrow \oint \left(\epsilon_0\vec{E} + \vec{P}\right) \cdot d\vec{A} = \iiint_\Omega \rho_f \ dV$$ Then $$\oint \epsilon_0\vec{E} \cdot d\vec{A} = \iiint_\Omega \rho_f \ dV - \oint \vec{P} \cdot d\vec{A}$$
Applying the divergence theorem and combining the right side integrals:
$$\oint \epsilon_0\vec{E} \cdot d\vec{A} = \iiint_\Omega \rho_f - \nabla \cdot \vec{P} \ dV $$
Finally, ##-\nabla \cdot \vec{P}## is what we call the bound charge density ##\rho_b##, so the integrand on the right side is ##\rho = \rho_f + \rho_b##, or the total charge density including both free and bound charges, so we're left with $$\oint \vec{E} \cdot d\vec{A} = \frac{1}{\epsilon_0}\iiint_\Omega \rho \ dV $$
What is bound charge density? Well, consider a conducting place in an electric field. Because of the applied external field, the atoms in the material form small dipoles, each of which cancels out the field from the adjacent dipole, except on the edges of the material. This surface charge constitutes the bound charge density. This, by the way, demonstrates the importance of always being aware of explicitly what it is that your variables represent when applying equations.
Think of ##\vec P## as "buried" inside ##\vec D##: $$\oint \vec D \cdot d \vec a = \int \rho_{free} dV \\ \oint (\varepsilon_0 \vec E + \vec P) \cdot d \vec a = \int \rho_{free} dV$$ Similarly ##\vec M## is "buried" inside ##\vec H##.Isaac0427 said:where are the P and M terms in some of maxwells equations, such as Gauss' law with the E field?
I get most of it, it's a little complicated with your wording, but I got the point.Charles Link said:@Isaac0427 Please read my post #16. Would enjoy your feedback.
Good. I wanted to show you an example (the dielectric sphere in a uniform electric field) that they might spend a week on in an advanced E&M class using Legendre Polynomials along with Poisson's equation for the electrostatic potential. I summarized the result in the example. When you see it in your coursework, you will already have had an introduction to it.Isaac0427 said:I get most of it, it's a little complicated with your wording, but I got the point.
Unfortunately they won't let me get that in my coursework for another 2 yearsCharles Link said:Good. I wanted to show you an example (the dielectric sphere in a uniform electric field) that they might spend a week on in an advanced E&M class using Legendre Polynomials along with Poisson's equation for the electrostatic potential. I summarized the result in the example. When you see it in your coursework, you will already have had an introduction to it.
Isaac0427 said:Unfortunately they won't let me get that in my coursework for another 2 years
Electromagnetism is a branch of physics that deals with the study of electric and magnetic fields and their interactions with each other. It is a fundamental force of nature that helps explain the behavior of charged particles and their motion.
Electric fields are created by stationary electric charges, while magnetic fields are produced by moving electric charges. Electric fields are responsible for the attraction or repulsion of charges, while magnetic fields cause charged particles to experience a force perpendicular to their direction of motion.
Electricity and magnetism are two sides of the same coin, known as electromagnetism. They are closely related, with electric fields influencing magnetic fields and vice versa. This relationship is described by Maxwell's equations, which unify the two phenomena into a single theory.
There are countless practical applications of electromagnetism, including generators, motors, transformers, and various electronic devices. Electromagnetic waves, such as radio waves, microwaves, and light, are used for communication and information transfer. Electromagnetic fields are also used in medical imaging and particle accelerators.
Electromagnetism plays a crucial role in the functioning of the universe. It is responsible for the formation of atoms, as well as the behavior of celestial objects such as planets, stars, and galaxies. The interaction between electric and magnetic fields also plays a role in the generation of cosmic rays and the structure of the universe on a large scale.