Electric field inside a spherical cavity inside a dielectric

[tanaygupta2000]

Homework Statement

Consider a spherical cavity in an infinite, homogenous and isotopic dielectric material of
permittivity ε. When placed in an external electric field E, the electric field inside the
cavity is ?

Relevant Equations

D = εE + P

 

[Charles Link]

I'm not sure the problem is phrased in an unambiguous way. Is the electric field $E$ inside the dielectric, or is this a dielectric slab that contains a cavity that is placed in an electric field $E$? Unlike a spherical dielectric in a uniform field, where the field inside the dielectric is constant, I don't know that this similar problem has an exact answer. [Edit: I see this does have a solution. See post 4.] @vanhees71 Might you be able to provide some insight?
Edit: See https://www.physicsforums.com/threa...n-dielectric-subject-to-uniform-field.257610/

 

[hutchphd]

If you can solve the polarized sphere in space and just subtract it from the uniformly filled space to make the hole. I think you get (a)

Incidentally the equation for D isn't correct

 

[Charles Link]

See https://www.physicsforums.com/threa...field-of-a-uniformly-polarized-sphere.877891/ post 7. The potentials will get an additional term $-Er \cos{\theta}$. The "link" in the "Edit" of my post 2 above has it right. I solved for $\sigma_{e}=-\sigma_{po}$ like it suggested. The calculation is non-trivial, but reasonably routine. $\\$ Note: $P_{r}(R)=\epsilon_o \chi E_{out \,r}(R)$, and $\sigma_p=-P_r (R)$.
$E_{out}=-\nabla V_{out}$, and $\chi =\frac{\epsilon}{\epsilon_o}- 1$.
Taking derivatives on the expression for the potential and taking the "r" component gives $(\frac{-2 \sigma_{po} R^3}{3 \epsilon_o R^3}-E) \chi \epsilon_o \cos{\theta}=\sigma_{po} \cos{\theta}$.
The Legendre method works for this. There may be other ways of doing it, but I don't know of any.

 

[Charles Link]

In the above, once you find $\sigma_{po}$, the electric field in the cavity $E_{total}$ is given by the very well known solution of $E_{total}=E+\frac{-\sigma_{po}}{3 \epsilon_o}$. $\\$ It should be noted in the above, (post 4), that the Legendre solution is an "ansatz" type solution, in which you make an educated guess, and you then show that the solution works. $\\$ Suggestion is to first work the problem of a uniformly polarized dielectric sphere using the Legendre method, and after that you should be ready for this more difficult problem.

 

[kuruman]

Although the Legendre method is the sure way to go, this is a multiple choice question where we are called upon to pick, not derive, the correct answer. Under the assumption that the correct answer is one of the 4, we can perhaps find physical arguments on the basis of which 3 choices can be declared incorrect. For example, in the limiting case $\epsilon=\epsilon_0$ is equivalent to replacing the dielectric with vacuum. In that case, the field everywhere must be the same and equal to the external field. This eliminates two of the choices which BTW are dimensionally incorrect. The third elimination can be achieved by considering what happens to the remaining expressions when $\epsilon>>\epsilon_0$.

 

[Delta2]

@kuruman thinking qualitatively the electric field inside a dielectric is weakened due to the polarization, can we say the same for the electric field inside the cavity?

 

[kuruman]

@kuruman thinking qualitatively the electric field inside a dielectric is weakened due to the polarization, can we say the same for the electric field inside the cavity?

Actually, I take back what I said about letting the dielectric constant become very large. This multiple choice question may be a derivative of the problem of finding the electric field inside a sphere placed in a uniform field $\vec E_0$. The answer to that is¹ $$\vec E=\frac{3\epsilon_0}{\epsilon +2\epsilon_0}\vec E_0.$$Here, all one has to do is swap the epsilons which gives one of the choices, the argument being that the epsilons are just constants particular to a given medium and if you swap the media leaving the external field alone, you must swap the constants.

___________________________________
¹ The uniform field with a dielectric sphere and with a cavity in a dielectric (plus more) are treated here.

 

[Charles Link]

@kuruman thinking qualitatively the electric field inside a dielectric is weakened due to the polarization, can we say the same for the electric field inside the cavity?

No=It's stronger inside the cavity.

 

[Vanadium 50]

Um...why are we providing complete solutions here?

 

[hutchphd]

No=It's stronger inside the cavity.

Yes. Surely stronger than the E field inside the material but how do you know stronger than E_outside? I don't see an easy argument to differentiate between (a) and (b) on that basis.

 

[Delta2]

Yes. Surely stronger than the E field inside the material but how do you know stronger than E_outside? I don't see an easy argument to differentiate between (a) and (b) on that basis.

Yes it seems there is no easy argument based on qualitative thinking that can lead to the conclusion that the E-field inside the cavity is stronger than $E_{outside}$. If someone finds one please tell !. In fact it seems a bit paradoxical to me, how the induced polarization of the dielectric material is such as to weaken the e-field inside the dielectric (fine so far) BUT the induced surface charge density at the surface of the cavity is such as to reinforce the electric field inside the cavity .

 

[kuruman]

Yes it seems there is no easy argument based on qualitative thinking that can lead to the conclusion that the E-field inside the cavity is stronger than $E_{outside}$. If someone finds one please tell !. In fact it seems a bit paradoxical to me, how the induced polarization of the dielectric material is such as to weaken the e-field inside the dielectric (fine so far) BUT the induced surface charge density at the surface of the cavity is such as to reinforce the electric field inside the cavity .

The argument is in the boundary conditions at the cavity surface. There are no free charges there so (a) the normal component of $\vec D$ must be continuous and (b) the tangential component of $\vec E$ must be continuous. From the continuity of normal components, $$\epsilon E_{out,n}=\epsilon_0 E_{in,n}~~\rightarrow~~E_{in,n}=\frac{\epsilon}{\epsilon_0}E_{out,n}$$From the continuity of tangential components,$$E_{in,t}=E_{out,t}$$Because $\epsilon > \epsilon_0,$ the normal component inside is greater than the normal component outside while the tangential components are equal. Therefore the magnitude of the field inside is greater than outside.

 

[Delta2]

thanks @kuruman I think that explains it all for me!

 

[Delta2]

sorry @kuruman i think again over your post #13 and i think that it proves that the electric field inside cavity is stronger than the electric field inside the dielectric. I am not sure if that's enough, cause if i interpret the problem correctly there is an external field $\mathbf{E_{outside}}$ which we assume is uniform at infinity (not sure about this interpretation) in which we bring a dielectric slab that has a spherical cavity in it. It is obvious that the electric field inside dielectric is smaller than $\mathbf{E_{outside}}$ and also smaller than the electric field inside cavity, but how we can prove that the electric field inside cavity is bigger than $\mathbf{E_{outside}}$? (if it is indeed bigger)
So we basically want to prove that $$\mathbf{E_{dielectric}}<\mathbf{E_{outside}}<\mathbf{E_{cavity}}$$, while it should be obvious that $$\mathbf{E_{dielectric}}<\mathbf{E_{outside}}$$ and i believe your post #13 proves that $$\mathbf{E_{dielectric}}<\mathbf{E_{cavity}}$$

 

[hutchphd]

Indeed that was the gist original question in #11.
But then I realized the the statement of the question is sort of ambiguous. The infinite dielectric fills the entire space. If it were a finite capacitor, the answer would depend upon how the dielectric was inserted...fixed free charge on plates or fixed potential difference.
I guess the most reasonable supposition is fixed charge but then (I believe) none of the answers given is correct. So I think the logic of @kuruman provides the only self-consistent solution.

 

[kuruman]

Indeed that was the gist original question in #11.
But then I realized the the statement of the question is sort of ambiguous. The infinite dielectric fills the entire space. If it were a finite capacitor, the answer would depend upon how the dielectric was inserted...fixed free charge on plates or fixed potential difference.
I guess the most reasonable supposition is fixed charge but then (I believe) none of the answers given is correct. So I think the logic of @kuruman provides the only self-consistent solution.

I agree 100%. The ambiguity was first pointed out by @Charles Link in #2. The question "When placed in an external electric field E, the electric field inside the cavity is ?" does not specify what is placed in the external field, just the cavity or the dielectric with the cavity. "Just the cavity" seems to be the author's intention because it provides one of the possible choices whereas "the dielectric with the cavity" does not.

In any case, we haven't heard from @tanaygupta2000 for 5 days so I will refrain from additional posts here until we do.