Point charge outside a conducting sphere: method of images

Using the method of images to find the electric field that results when a charged particle is placed outside a conducting sphere. At the end, we consider how this relates to the force on the particle in the cases of both grounded and isolated conductors.
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 #electrostatics #electricfield #electricity #charge #sphere #conductor #equipotential #potential #methodofimages #uniqueness #uniquenesstheorem #gravity #gravitation #math #science #education

[@MissPiggyM976]

Another very good e.m. video! Thanks!

[@AbhishekSingh-pg4lc]

where is the distance taken from

[@tharuniist-jee-hz8kc]

From the origin

[@rhvc2734]

If the conductor had some non zero surface charge density, then what approach should I take?

[@DrBenYelverton]

If there's a net charge on the conductor, start with the solution for the non-grounded neutral sphere, but include yet another image charge at the centre of the sphere, with a charge equal to the total charge on the conductor.

[@rhvc2734]

@@DrBenYelverton ok thankyou🙏

[@mra8554]

but if your original source is inside the sphere, then an image charge cannot be placed in the center right?
@@DrBenYelverton

[@DrBenYelverton]

@@mra8554 You're right, we can't place image charges in the region we're solving in - they'd introduce singularities that shouldn't be there. In fact, I don't think you'd need an extra image charge at all if you're solving inside the sphere. You can view the charge distribution on the sphere as a superposition of the original uniform charge density, plus an extra charge density induced by the point charge. The uniform component doesn't actually affect the field inside, which you can see from Gauss' law (or equivalently Newton's shell theorem), so the conclusion is that adding a nonzero surface charge to the conductor doesn't make any difference inside.

[@somabose7667]

if it was just a non grounded neutral conducting sphere and a point charge, how to approach the problem?

[@DrBenYelverton]

This is discussed at the end of the video, from around 15:00 onwards.

[@somabose7667]

​@@DrBenYelverton but the thing is when I computed th surface charge density I am getting a variable factor (1/a- cos theta)^(3/2) and I am not being able include the fact that one half has negative charge density and other has positive.... I have only been able to figure out the nature of variation

[@pestermaster7617]

I have to calculate point charge, near infinite grounded tube with a set radius. Is it even possible without doing it numericaly? I think it would look similary to infinite line of charge near a tube, but the charge density would change along the tube. Is it doable?

[@DrBenYelverton]

It's not possible to solve exactly - it seems that approximate solutions do exist but the maths is very involved, even in the symmetrical case where the charge is at the centre of the cylinder! See e.g. arxiv.org/abs/2001.10651

[@pestermaster7617]

@@DrBenYelverton thank you very much!

[@thayhungvalee]

If point charge inside a conducting sphere, then what approach should I take?

[@DrBenYelverton]

In the video we interpreted the charge outside as "real" and the charge inside as the image. But, we're free to interpret it the other way around, with the charge inside being "real" and the charge outside being the image. That means the solution for the case you described is the same as in the video. So for a charge Q at distance d, whether d > r or d < r, the image charge will be at a distance r²/d with a charge q = -(r/d)Q.

[@baconka]

@@DrBenYelverton Wouldn’t the point charge inside the conducting sphere produce electric field and breaks the electrostatic equilibrium of the sphere? Can we still solve the problem using the “inverse” method of images you’ve mentioned above? I tried to imagine the case that the point charge is in a cavity that covered the charge so that the sphere will still maintain electrostatic equilibrium. But it seems complicated how method of image will be used in this case😢(I can’t define the boundaries due to the cavity). Is there any misunderstanding and what approach can I take?And great video professor! I really enjoy your explanation.