Help with Maxwell’s equation in integral form

In summary, the electric field due to a cylindrical charge distribution using Gauss' law is given by: the flux of the induction, Q_{V}, is given by: \vec{D}=\frac{\rho_{0}(e-1)e^{-1}}{2r} a_{r}
  • #1
robert25pl
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0
Electric field due to a cylindrical charge distribution using Gauss' law.
Charge is distributed with density [tex]\rho_{0}e^{-r^{2}}[/tex] C/m^3 in cylindrical region r < 1. Find D (displacement flux density vector) everywhere.

I did used this equation
[tex]\int_{s}D\cdot\,dS=\int_{V}\rho\*d\upsilon[/tex]

Since this is a cylindrical charge distribiution I used Gaussian surface in the shape of a cylinder.

[tex]\int_{s}D\cdot\,dS=\rho\*l[/tex]

So if I understand good the D=0 inside cylinder. therefore r>R is valid.
The surface area is [tex]2\pi\*rL[/tex].

I'm having a problem to set up the the equation or I'm doing everything wrong?
Thanks for any help and recommendation.
 
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  • #2
1.U need to evaluate the charge...What's the charge in the cylinder...?


Daniel.
 
  • #3
[tex]\int_{s}D\cdot\,dS=\int_{V}\rho\*d\upsilon[/tex]
So fom this I need to evaluate volume integral?

[tex]\int_{s}D\cdot\,dS=Q_{V}[/tex]

[tex]Q_{V}=\int_{V}\rho_{0}e^{-r^{2}}[/tex]

But what limits should I used for x,y,z (0 and 1 for all three)?
Thanks
 
  • #4
Nope,you need the cilindrical coordinates [tex] r,\varphi,z [/tex].What's the volume element in cilindrical coordinates...?

Daniel.
 
  • #5
[tex]Q_{V}=\int_{V}\rho_{0}e^{-r^{2}}[/tex]

[tex]Q_{V}=\int_{r=0}^{r}\int_{\phi=0}^{2\pi}\int_{z=0}^{l}\rho_{0}e^{-r^{2}}dr\,d\phi\,dz[/tex]

Is that correct?
 
  • #6
U need another "r" in the volume element.And the cilinder has radius 1 (see text of the problem)...

Daniel.
 
  • #7
[tex]Q_{V}=\int_{r=0}^{1}\int_{\phi=0}^{2\pi}\int_{z=0}^{l}\rho_{0}e^{-r^{2}}r dr\,d\phi\,dz[/tex]

That is not easy integration, but I think I got it.

[tex]Q_{V}=l\pi\rho_{0}(e-1)e^{-1}}[/tex]

Is this is correct what would be next step?
 
  • #8
It is correct.Now u have to apply Guass' theorem which gives the flux of the induction [itex] \vec{D} [/itex]...

Daniel.
 
  • #9
if I understand well D depend on r only, so:

[tex]\int_{s}D\cdot\,dS=\int_{\phi=0}^{2\pi}\int_{z=0}^{l}D_{r}(r)a_{r}\cdot{r}\,d\phi\,dz\,a_{r}=[/tex]

[tex]=2\pi\,rlD_{r}(r)[/tex]

And this I should compare to Qv and find D, right?
 
  • #10
It's the same cyclinder of radius unity...That "r" is 1...

Daniel.
 
  • #11
So this is my D

[tex]D=\frac{\rho_{0}(e-1)e^{-1}}{2r} a_{r}[/tex]

I really appreciate your help
 

1. What are Maxwell’s equations in integral form?

Maxwell's equations in integral form are a set of four equations that describe the relationship between electric and magnetic fields, and how they interact with charged particles. These equations are fundamental to understanding electromagnetism and are used in various fields such as optics, electronics, and telecommunications.

2. What is the significance of Maxwell’s equations in integral form?

Maxwell's equations in integral form are significant because they provide a complete and elegant description of electromagnetic phenomena. They are also crucial in the development of technologies that rely on electromagnetic fields, such as radio, television, and cell phones.

3. How do you use Maxwell’s equations in integral form?

Maxwell's equations in integral form can be used to solve problems involving electric and magnetic fields. They can also be used to predict the behavior of electromagnetic waves and calculate properties such as their speed and wavelength.

4. What are the advantages of using Maxwell’s equations in integral form?

The integral form of Maxwell's equations allows for a more concise and elegant representation of electromagnetic phenomena compared to the differential form. It also makes it easier to apply them in practical situations, such as in the design of electronic devices.

5. Are there any practical applications of Maxwell’s equations in integral form?

Yes, there are numerous practical applications of Maxwell's equations in integral form. They are used in the design of antennas, microwaves, and other electronic devices. They also play a crucial role in the development of technologies such as radar, satellite communication, and medical imaging.

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