How electric potential boundary condition works

[tomasg]

Homework Statement

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Inside a sperical dielectric mass there is a electric dipole on the center of the sphere. The sphere has radius a. This dieletric sphere is inside and on the center of a conductive spherical shell of radius b. The problem asks to find the potentials and then the electric fields in every region, inside the dielectric sphere, the space between the sphere and the shell and outside the shell.

Homework Equations

Its given that p=p0*z (the dipole looks towards +z )

The Attempt at a Solution

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Now, i have written all the potentials (the solutions of laplace) but i noticed that i haven't fully understood one boundary condition for the electric potential. The one that says ε2(∂Vout/∂r)-ε1(∂V/∂r)=-σ(θ)/ε0.
The problem doesn't say anything about the charge of the shell, so i suppose is zero. So my question is this, does the σ(θ) of the above condition refers to the induced charge density (which would not be zero in this example i think) or the charge density of the shell alone?

 

[Govind Lal Sidhardh]

well I'm not an expert so pardon me if I'm wrong...But the above equation refers to the discontinuity of the electric field in the boundaries of two media,which is derived from Gauss' law.And the charge density in Gauss' law is the net charge density(due to induction too).So I think the charge density refers to the net charge density,which in this case is the induce charge density...

 

[tomasg]

Thanks for the reply. I believe this is true too. Atleast that's what i have understand after reading more caerefully griffith's book

 

[TSny]

ε2(∂Vout/∂r)-ε1(∂V/∂r)=-σ(θ)/ε0.

This equation does not look quite right. The left side gives the change in the radial component of D (not E). The right hand side should then be -σ_free(θ) without any ε0.

σ_free is the free charge density on the boundary surface between the two dielectric materials.

If you wrote the left hand side without the ε2 and ε1 as ∂Vout/∂r- ∂V/∂r, then you now have radial components of E. Then the right hand side would be -σ(θ)/ε0 and σ(θ) would be the total surface charge density (free plus bound).

I have an older 3rd edition of Griffiths. In this edition, there is a section 4.3.3 called "Boundary Conditions". Here you find the equations

D^⊥_above - D^⊥_below = σ_f where σ_f is free charge density.

and

E^⊥_above - E^⊥_below = σ/ε0 where σ is total charge density.

 

[tomasg]

This equation does not look quite right. The left side gives the change in the radial component of D (not E). The right hand side should then be -σ_free(θ) without any ε0.

σ_free is the free charge density on the boundary surface between the two dielectric materials.

If you wrote the left hand side without the ε2 and ε1 as ∂Vout/∂r- ∂V/∂r, then you now have radial components of E. Then the right hand side would be -σ(θ)/ε0 and σ(θ) would be the total surface charge density (free plus bound).

I have an older 3rd edition of Griffiths. In this edition, there is a section 4.3.3 called "Boundary Conditions". Here you find the equations

D^⊥_above - D^⊥_below = σ_f where σ_f is free charge density.

and

E^⊥_above - E^⊥_below = σ/ε0 where σ is total charge density.

yes i should not put the ε0 there. Thank you so much sir for the answer. It finally makes sense to me. And as the user above said, the E^⊥_above - E^⊥_below = σ/ε0= (σf+σb)/ε0 <--- in this equation the σf could be the induced charge density or the charge density we created in the conductor (or both). Right?

 

[TSny]

And as the user above said, the E^⊥_above - E^⊥_below = σ/ε0= (σf+σb)/ε0 <--- in this equation the σf could be the induced charge density or the charge density we created in the conductor (or both). Right?

I'm not sure I'm understanding your question. In your problem, the inner surface of the conducting sphere will have a free charge density σ_f which is the charge density induced on the inner surface of the spherical conductor by the dielectric sphere with the dipole. So if you are applying the equation to the inner surface of the conductor, the σ in σ/ε0 would be this σ_f.