Point charge inside Dielectric Sphere embedded in Dielectric Slab

In summary, according to the conversation, one should calculate potential using D-E/4pi and Phi, and allow for potential to be continuous at the interface. However, the potential must be continuous up to a constant, which can be determined by calculating the polarization. Additionally, one should account for the potential due to the point charge when calculating potential.
  • #1
OsCiLL8
3
0
I've been working on this for a little while now (in CGS units), and am not really sure where I've gone wrong at in calculating the potential, so I've come here! Here is the problem:

What is the potential caused by placing a point charge Q at the center of a dielectric sphere ([tex]\epsilon[/tex]2), radius R, that is embedded inside some other infinite slab of dielectric ([tex]\epsilon[/tex]1)?

Here's what I've determined so far:
D(r) = Q/r2
E(r<R) = Q/[tex]\epsilon[/tex]2*r2
E(r>R) = Q/[tex]\epsilon[/tex]1*r2

So, letting P = (D-E)/4[tex]\pi[/tex] , I've found

[tex]\Phi[/tex](r<R) = Q/r + ([tex]\epsilon[/tex]2-1)*Q/(3*[tex]\epsilon[/tex]2*r)

[tex]\Phi[/tex](r>R) = Q/r + ([tex]\epsilon[/tex]1-1)*Q/(3*[tex]\epsilon[/tex]1*r)


My question is, shouldn't I have the option of allowing the potential to be continuous at the interface?? Have I left out some surface charge polarization or something?
 
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  • #2
OsCiLL8 said:
I've been working on this for a little while now (in CGS units), and am not really sure where I've gone wrong at in calculating the potential, so I've come here! Here is the problem:

What is the potential caused by placing a point charge Q at the center of a dielectric sphere ([tex]\epsilon[/tex]2), radius R, that is embedded inside some other infinite slab of dielectric ([tex]\epsilon[/tex]1)?

Here's what I've determined so far:
D(r) = Q/r2
E(r<R) = Q/[tex]\epsilon[/tex]2*r2
E(r>R) = Q/[tex]\epsilon[/tex]1*r2

Okay, so far so good...

So, letting P = (D-E)/4[tex]\pi[/tex] , I've found

[tex]\Phi[/tex](r<R) = Q/r + ([tex]\epsilon[/tex]2-1)*Q/(3*[tex]\epsilon[/tex]2*r)

[tex]\Phi[/tex](r>R) = Q/r + ([tex]\epsilon[/tex]1-1)*Q/(3*[tex]\epsilon[/tex]1*r)

Why are you calculating the polarization, and how did you go from the polarization to the potential?


My question is, shouldn't I have the option of allowing the potential to be continuous at the interface??

"Option" is a poor choice of words, the potential must be continuous everywhere...the fact that yours is not should be a dead giveaway that you've done something wrong.
 
  • #3
Calculating the polarization allows me to determine the contribution to the potential from the polarization surface charge density and polarization volume charge density.

I think your incorrect about the need for a continuous potential. The parallel E component and the perpendicular D componenent have to be continuous, implying that the potentials at the boundary be equal UP TO a constant. Setting the constant equal to zero makes the potential continuous, while setting the constant equal to a nonzero number implies that there is some work function required to go from one dielectric to the other.
 
  • #4
OsCiLL8 said:
Calculating the polarization allows me to determine the contribution to the potential from the polarization surface charge density and polarization volume charge density.

Okay, but you will also need to account for the potential due to the point charge, and you have made an error somewhere. If you post your calculations, I can point it out to you.

Alternatively, you can save yourself from the hassle of this method altogether by just using the definition of potential:

[tex]\textbf{E}=-\mathbf{\nabla}V\Longleftrightarrow V(\textbf{r})=\int_{\mathcal{O}}^{\textbf{r}}\textbf{E}\cdot d\mathbf{l}[/tex]

I think your incorrect about the need for a continuous potential. The parallel E component and the perpendicular D componenent have to be continuous, implying that the potentials at the boundary be equal UP TO a constant. Setting the constant equal to zero makes the potential continuous, while setting the constant equal to a nonzero number implies that there is some work function required to go from one dielectric to the other.

No, the potential must be continuous everywhere...What would the force be on a point charge located at a discontinuity in potential?
 
  • #5
The integral is a continuous function of the upper limit. Integrate E(r) from the place where the potential is chosen 0 (that is infinity) to r.

ehild
 

What is a "point charge"?

A point charge is a theoretical concept used in physics to represent a particle with a significant amount of charge concentrated at a single point. It is often used to simplify calculations and understand the behavior of charged particles in electric fields.

What is a "dielectric sphere"?

A dielectric sphere is a spherical object made of a material that can be polarized by an electric field. This polarization results in an induced electric dipole moment, which affects the behavior of the electric field around the sphere.

What is a "dielectric slab"?

A dielectric slab is a thin, flat object made of a material that can be polarized by an electric field. It behaves similarly to a dielectric sphere, but its shape and dimensions may affect the behavior of the electric field in more complex ways.

What happens when a point charge is placed inside a dielectric sphere embedded in a dielectric slab?

The electric field around the point charge will be affected by the presence of the dielectric sphere and slab. The polarization of the dielectric materials will cause a distortion in the electric field lines, and the resulting field will be weaker inside the sphere and slab compared to the outside.

How do you calculate the electric potential and field inside and outside a dielectric sphere embedded in a dielectric slab?

To calculate the electric potential and field in this scenario, you need to use the appropriate equations and boundary conditions for the electric potential and field of a point charge in the presence of dielectric materials. These calculations may involve solving differential equations and applying appropriate boundary conditions to determine the electric potential and field at different points in space.

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