What is the reason behind the charge residing on the surface of the conductor having the same magnitude as that of image charge?

Question

Lately, I'm studying the method of images especially to mention the renowned example of "A point charge near a conducting sphere". So, we have a grounded conducting sphere & a point charge $\displaystyle{q}$ . The image charge, after computation, is $\displaystyle{q'} = -\frac{a}{b}\displaystyle{q}$ . Then easily we can find the field outside by using superposition principle & also find the force on $\displaystyle{q}$ solely by the field of $\displaystyle{q'}$. All is well; but then when I integrate over the whole surface, I do get the total charge on the surface of the conducting sphere is $\displaystyle{q'}$. As marked by Feynman

[...]the total induced charge is $\displaystyle{q'}$, as it should be.

I didn't get at the physical reasoning behind getting the same magnitude of charge as that of image charge. The image charge was located at $\dfrac{a^2}{b}$ w.r.t. the center of the sphere while the surface is located at $a$. So, the distance between image charge & $q$ is $\left(b - \dfrac{a^2}{b}\right)$, while that of the surface & $q$ is $(b - a)$; so though the magnitude of the charge on the surface is same as that of the image charge, the distance is not the same. So, how can the fields be same? In order to have the zero potential, charge of $\displaystyle{q'}$ must be located at a distance of $\dfrac{a^2}{b}$ from the center of the sphere. If $q'$ be situated at some other position like at the surface which is not at $\dfrac{a^2}{b}$ but at $a$, then how can there be same field, after all the distance has been changed between $q$ & $q'$? I'm not understanding, thus why the charge at the surface is $\displaystyle{q'}$. Can anyone please explain me the reason??

Answer 1

Imagine two situations.

Situation one, you have a point charge of charge $q'$ at the image charge location.

Situation two, you have some charge distributed on the surface of the conductor.

In both cases you have the exact same electric field on the surface right outside the conductor. So you have the same electric flux there.

But electric flux is proportional to the charge enclosed. They have the same flux so the same charge enclosed. The one with an actual charge at the image charge location has a charge enclosed of $q'$ so that is the charge enclosed for the conductor as well.

So the net charge on the surface of the conductor must be $q'$ the same as the total charge of the image charge.

There could be regions where the surface charge density on the surface is negative and regions where it is positive and only the net total will equal $q'$ and in general the charge will be spread out throughout the surface not just located at one point.

If the field lines are coming into the conductor at a location A there is a negative surface charge at location A, for instance there could be an excess of electrons at location A. If there field lines are coming out of the conductor at a location B then there is a positive surface charge at location B, for instance there could be fewer electrons at location B than protons at location B.

If a sphere started out neutral and isolated and ungrounded then it can't have a net charge so you can place an equal and opposite amount of charge equally distributed across the whole sphere. The sum of those surface charge densities is then then actual net surface charge density. As for how the charge gets there all of it is in reaction to the external charge. But in many cases it fill flow through the conductor from one end of the surface to the other. If the external charge comes in slowly enough then it can flow through the body of the conductor in a steady way where almost no charge buildup forms in the body of the conductor.

Grounded and ungrounded try to change the potential of the conductor by putting whatever charge is needed uniformly on the sphere. The amount needed depends on how far away the external charge is; when the external charge is greater you have to have more charge uniformly added to the surface in order to counter that very strong potential the external charge makes right at the surface.