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

AI Thread Summary
The discussion centers on calculating the electric displacement field D2 across a dielectric boundary defined by the equation 4x + 3y = 10 m, with the first region being free space. The user initially arrives at a different value for D2 than what is provided in the textbook, leading to confusion. It is suggested that the discrepancy arises from an error in the textbook regarding the relative permittivity of the second medium, which should be 0.4 instead of the stated 2.5. Following this correction, the calculations align with the expected answer. This highlights the importance of verifying given parameters in physics problems.
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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" ), 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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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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