Find Flux Density On One Side of Dielectric Boundary Given Boundary Conditions

In summary: This is the video: https://www.youtube.com/watch?v=6kQ5p1IjVz0And the answer in the video is 0.416ax - 1.888ay + 2.6az nC/m2 as the OP says the answer in the back of the book is.
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Bizkit
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Homework Statement


A dielectric interface is defined as 4x + 3y = 10 m. The region including the origin is free space, where D1 = 2ax - 4ay + 6.5az nC/m2. In the other region, εr2 = 2.5. Find D2 given the previous conditions.


Homework Equations


an12 = ± grad(f)/|grad(f)|

D2n = D1n = an(D1 · an)

D1t = D1 - D1n

ε = ε0εr

D2t = (D1t)(ε2)/ε1

D2 = D2n + D2t


The Attempt at a Solution


f = 4x + 3y - 10 = 0

an12 = ± grad(4x + 3y - 10)/|grad(4x + 3y - 10)| = ± (4ax + 3ay)/5 = ± (.8ax + .6ay) Since the vector points in the positive x and y directions, I choose the plus sign to get: an12 = .8ax + .6ay

D2n = D1n = (.8ax + .6ay)((2ax - 4ay + 6.5az) · (.8ax + .6ay)) nC/m2 = (.8ax + .6ay)(-.8) nC/m2 = -.64ax - .48ay nC/m2

D1t = (2ax - 4ay + 6.5az nC/m2) - (-.64ax - .48ay nC/m2) = 2.64ax - 3.52ay + 6.5az nC/m2

ε1 = ε0εr1 = ε0 = 8.854 pF/m (since the region is free space)

ε2 = ε0εr2 = (8.854 pF/m)(2.5) = 22.135 pF/m

D2t = (2.64ax - 3.52ay + 6.5az nC/m2)(22.135 pF/m)/(8.854 pF/m) = (2.64ax - 3.52ay + 6.5az nC/m2)(2.5) = 6.6ax - 8.8ay + 16.25az nC/m2

D2 = (-.64ax - .48ay nC/m2) + (6.6ax - 8.8ay + 16.25az nC/m2) = 5.96ax - 9.28ay + 16.25az nC/m2

The answer in the back of the book, however, is given as D2 = .416ax - 1.888ay + 2.6az nC/m2, which is completely different than what I got. I'm not sure where I went wrong. I followed one of my teacher's examples that he has posted (you can find it http://montoya.sdsmt.edu/ee381/examples/tilt_dielectric_boundary.pdf" [Broken]), which is very similar to this question, but I still come up with the wrong answer. Can someone please show me where I went wrong? Thanks.
 
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  • #2
This question is 12+ years old at the time of answering, but maybe the following will be helpful to someone.

It appears that the question has a mistake. The given value of relative permittivity for medium 2 is wrong.

If we use ##\epsilon_{r2} = 0.4## instead of the stated value (##\epsilon_{r2} = 2.5##) then using the OP’s method gives the ‘official’ answer exactly.

I discovered this from a YouTube video where an (almost) identical problem is solved. The presenter explicitly changes the stated value of ##\epsilon_{r2}## early on but doesn't immediately say why. But near the end of the video, the presenter explains that there is a mistake with the textbook’s given value of ##\epsilon_{r2}## which is why she changed it to 0.4,.

 
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What is flux density?

Flux density, also known as magnetic flux density or magnetic induction, is a measure of the strength of a magnetic field. It is a vector quantity, meaning it has both magnitude and direction.

What are boundary conditions?

Boundary conditions refer to the set of conditions that must be satisfied at the interface between two different materials or regions. These conditions are necessary for solving electromagnetic problems and determining the behavior of the fields at the boundary.

What is a dielectric boundary?

A dielectric boundary is the interface between two materials with different dielectric properties. A dielectric material is one that does not conduct electricity, but can store electrical energy. Examples include glass, rubber, and plastic.

How do boundary conditions affect flux density?

Boundary conditions play a crucial role in determining the behavior of the flux density at the interface between two materials. They can affect the direction and magnitude of the flux density, as well as any changes in the flux density as it crosses the boundary.

How do I find the flux density on one side of a dielectric boundary given boundary conditions?

The flux density at a dielectric boundary can be calculated using Maxwell's equations and the given boundary conditions. The exact method for solving this problem will depend on the specific conditions provided, but it typically involves using mathematical techniques such as boundary element method or finite element method.

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