Physics 46 Maxwell's Equations (14 of 30) Differential Form of Gauss' Law: 6

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In this video I will give an example (4) of the divergence of a vector of a Gaussian sphere.

[@raheome]

You are one of the best professors I have ever encountered. I can never thank you enough!

[@MichelvanBiezen]

raheome, Thank you for the feedback. I am glad you are enjoying these videos.

[@dipayan4264]

Zero dislikes..... that shows the clarity of interpretations......

[@maheshwaritanwar3012]

This made so much sense!

[@MichelvanBiezen]

Glad it did.

[@galaxygeckocreations]

Hi Michael! I am a Physics/Environmental Science double-major going into my senior year of college, and I just want to say thank you so much for posting these videos. I did not get a chance to take the upper-level E&M course at my school, so I never went over Maxwell's equations in detail. But from what I can tell, they seem important to know for grad school, so I've decided to learn them on my own before I start my last year of undergrad. These videos are a godsend, so incredibly helpful - you explain everything very well and I'm really understanding it so far!

[@MichelvanBiezen]

It is good to know the "basics" before starting grad school. Good luck with your studies and keep on going.

[@khaileng3020]

Actually the charge density is not divided by Ro because the gaussian surface is somewhere at the point inside the sphere insulator , it should be Rg ,if the gaussian surface is at the outside of the sphere then Ro , am i right?

[@CatsBirds2010]

Very nice! Thanks.

[@MichelvanBiezen]

Thank you.

[@altuber99_athlete]

This was a beautiful proof.

[@lamisselkhattouti1568]

first of all I want to thank you so much for these videos, I've a question please : for the Q in the gauss's cercle it's 3/4 pi r^3 that's what we were doing in class so why did we add the R sub knot here!

[@ianhnizdo5864]

Your equation is similar but different from the proof of the electric field for a sphere in your series on Gauss's law. There you found the field was kQr/(R^3), here its 3kQ/(R^3). I'm curious why that's the case unless the two equations are equivalent. If that is the case how are they equivalent?

[@FirstGradeCalculus]

Great lectures! Thank you very much.

[@thomasdiprima2629]

Based on your explanation, that would mean that the divergence of E would be the same at any point along the radius. Please explain.

[@MichelvanBiezen]

These videos explain the meaning of divergence: CALCULUS 3 CH 8 DIVERGENCE AND CURL

[@altuber99_athlete]

It indeed is. The E field is the same for each point in the same radius, so the divergence of E is also the same fot any point in that radius.

[@HumphreyBriggs]

If divergence of the electric field is how it is changing then how does this differ from just differentiating it with respect to r? Is it because it is a vector and we may need to know how it changes in 3 directions? Thanks

[@MichelvanBiezen]

+HumphreyBriggs That is correct. The divergence will show you how it changes in any direction.

[@altuber99_athlete]

Yes. Since it's a 3D vector, we need to differentiate in each dimension (3 in total). And that's the definition of the divergence (in Cartesian coordinates).

[@solsticetwo3476]

What happen if r >Ro? Seems that field density E(r) increases with r. Is that possible? Also, you proved the identity of the differential form, but still is missing the insights, the meaning of that equation.

[@AngelAlita84]

how comes epsilon sub not just moves easily from power to power? you take it from denominator and add to the numerator without decreasing the power? can you explain it please?

[@MichelvanBiezen]

k = 1 / (4 x pi x epsilon sub knot)

[@altuber99_athlete]

As Michel answered you 6 months ago, he *didn't* move ϵ_0 from the denominator to the numerator, no. What he did was to identify that the inverse of ϵ_0, also multiplied by 4 and by π (i.e. 1/4πϵ_0), is equal to k.

[@AlanMedina314]

You Rock...

[@biggbuck9535]

Does anyone know where the (1/r²) and the r² around the partial derivative came from?

[@MichelvanBiezen]

That is part of finding the dV in spherical coordinates. You can find how that is done in this playlist: PHYSICS 67 ADVANCED ELECTRICITY & MAGNETISM

[@MathPhysicsFunwithGus]

@@MichelvanBiezen I'm not sure if I found the right video or not under that playlist but it doesn't really explain to me why you take 1/r^2 and such of the partial derivatives if you could explain that further please