Bound Charge at Dielectric Interface

Here is a question sent to me recently:

I will look at part (b). Let ϵ1 \epsilon_1 and ϵ2 \epsilon_2 be the relative electric permittivity values. The field continuity equation is (where the electric fields are perpendicular to the interface boundary):

ϵ1E1=ϵ2E2 \epsilon_1 E_1 = \epsilon_2 E_2

Relate the voltage across the capacitor to the electric fields:

V=E1d1+E2d2=E1(d1+ϵ1ϵ2d2)E1=Vϵ2d1ϵ2+d2ϵ1E2=Vϵ1d1ϵ2+d2ϵ1 V = E_1 d_1 + E_2 d_2 = E_1 \Big(d_1 + \frac{\epsilon_1}{\epsilon_2} d_2 \Big) \\ E_1 = \frac{V \epsilon_2}{d_1 \epsilon_2 + d_2 \epsilon_1} \\ E_2 = \frac{V \epsilon_1}{d_1 \epsilon_2 + d_2 \epsilon_1}

Consider the Gaussian surface shown to the right within the dielectric. The formula for bound charge at a dielectric interface is similar to the formula for Gauss's Law, but with a slight variation. Surfaces S2 S_2 and S1 S_1 are the top and bottom Gaussian sub-surfaces.

Qb=(ϵ11)ϵ0S1E1dS1+(ϵ21)ϵ0S2E2dS2 Q_b = (\epsilon_1 - 1) \epsilon_0 \int \int_{S1} \vec{E}_1 \cdot \vec{dS_1} + (\epsilon_2 - 1) \epsilon_0 \int \int_{S2} \vec{E}_2 \cdot \vec{dS_2}

Note that the normal vector for S1 S_1 points in the opposite direction as the electric field, and the normal vector for S2 S_2 points in the same direction as the electric field. Let these surfaces both have area S S . The equation simplifies to:

Qb=(ϵ11)ϵ0(E1S)+(ϵ21)ϵ0(E2S) Q_b = (\epsilon_1 - 1) \epsilon_0 (- E_1 S) + (\epsilon_2 - 1) \epsilon_0 (E_2 S)

Applying the field continuity relationship makes the relative permittivity terms disappear.

Qb=ϵ0(E1S)ϵ0(E2S) Q_b = \epsilon_0 (E_1 S) - \epsilon_0 (E_2 S)

Dividing both sides by S S gives the bound charge density:

σ=ϵ0E1ϵ0E2=ϵ0(Vϵ2Vϵ1d1ϵ2+d2ϵ1) \sigma' = \epsilon_0 E_1 - \epsilon_0 E_2 = \epsilon_0 \Big( \frac{V \epsilon_2 - V \epsilon_1}{d_1 \epsilon_2 + d_2 \epsilon_1} \Big)

Note by Steven Chase
5 months, 2 weeks ago

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@Steven Chase Thankyou sir

A Former Brilliant Member - 5 months, 2 weeks ago

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That's impressive. You found it ten seconds after I uploaded it

Steven Chase - 5 months, 2 weeks ago

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@Steven Chase Sir can you please post a RLC problem . I think due to some reason you are upset with me. If yes then sorry. If no, then why you don't reply me?

A Former Brilliant Member - 5 months, 1 week ago

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There's a new one up now. I usually don't know when I'm going to post another one until just before I post it.

Steven Chase - 5 months, 1 week ago

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@Steven Chase Hel20Hel^{2}0 I was again solving this question because I was doing revision.
And i just got stuck in your first line. I have forgot the concept little bit.
That line ϵ1E1=ϵ2E2\epsilon_{1}E_{1}=\epsilon_{2}E_{2}
Just give me the proof of this line. Thanks in advance.
Hope I am not disturbing you.

Talulah Riley - 3 months, 3 weeks ago

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In the absence of free charge, the permittivity-weighted normal components of the electric field are continuous at a dielectric interface. Here is a further reference:

https://en.wikipedia.org/wiki/Interfaceconditionsforelectromagneticfields

Steven Chase - 3 months, 3 weeks ago

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@Steven Chase Hello, hope you are doing well
The formula which you have used for QbQ_{b} is it a original formula for finding bound charges ?

Talulah Riley - 2 months, 2 weeks ago

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