Bound charge density: why does ρ = -∇⋅P?

Proving that the bound charge density in a dielectric is minus the divergence of the polarisation, and gaining some physical intuition as to why this should be the case.
Bound surface charge: • Polarisation and surfa...
About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
My website: benyelverton.com/
#physics #mathematics #electromagnetism #boundcharge #charge #chargedensity #electriccharge #dielectric #electricfield #electricdipole #dipole #dipolemoment #dielectrics #insulator #materials #polarisation #polarization #vector #electrostatics #maths #math #science #education

[@amittksingh]

Thankyou for the last part where you explained on physical grounds why there will be a negative charge.

[@DrBenYelverton]

No problem, always interesting to try to understand equations intuitively!

[@TheJara123]

Interesting presentation, man, thank you

[@DrBenYelverton]

Thanks, glad to hear it! Will be posting some more videos on dielectrics soon.

[@mxminecraft9410]

Sir can you please make a video on all the mathematics or mathematical tools we require to study physics at different levels .

[@DrBenYelverton]

That's a big topic, I'd need to spend some time thinking about how best to approach that!

[@mxminecraft9410]

@@DrBenYelverton ok sir

[@douglasstrother6584]

Calculus, Linear Algebra and Differential Equations will take you far.

[@mingmiao364]

Very intuitive! Can we derive it using Gauss’s divergence theorem for an arbitrary surface?

[@DrBenYelverton]

We can indeed - see e.g. the Feynman Lectures, Vol 2, Chapter 10, for details! The method I used here is ultimately equivalent though, since proving the divergence theorem usually involves splitting the volume into small cube-shaped elements.

[@mingmiao364]

@@DrBenYelverton wow thanks for the reference! Will check it out!

[@DrBenYelverton]

The Feynman Lectures are absolutely my favourite Physics resource - very helpful for understanding things intuitively! I haven't read them all yet but am gradually working my way through them. All freely available via the official website too.

[@mingmiao364]

3:05 "By conservation of charge..." I have a question: why does the conservation of charge implies that the charge q_y(x,y,z) on the right face of the cube is balanced out completely by the charge on the left face, rather than by the charge on the other five faces (left, front, back, upper, lower) combined? Perhaps I'm missing something

[@DrBenYelverton]

Good point, the conservation of charge alone is not enough to imply that. However, we know that the effect of each component of the applied field is to pull positive charges in one direction and negative charges in the exact opposite direction, from which the result follows.

[@mingmiao364]

That makes sense, thanks!@@DrBenYelvertonDoes this argument still stand if the material is anisotropic tho?

[@DrBenYelverton]

@@mingmiao364 Yes, because there will still be a dipole induced in some direction, which can always be thought of as the sum of three orthogonal dipoles perpendicular to the faces of the cube. The only difference would be that P is no longer parallel to E, but that doesn't affect the logic as our derivation depends only on P, not E.

[@mingmiao364]

@@DrBenYelverton Right! I’m still used to always think in terms of E and that’s how I got confused! Great explanation, thank you!