- #1
TonyG
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OK, for fun I was solving a simple electrostatics problem, where a charge q is sitting some distance "x" from a grounded conducting sphere.
You can use the method of images for this, giving you an image charge q' located at a distance x' from the center of the sphere. The answer I get is:
q' = -qa/x
x' = a^2/x
(this doesn't really matter for my question)
Anyway, solving for the total charge induced on the sphere (by evaluating E at r=a, giving sigma, then integrating over the area of the sphere), it took pages and pages of grinding thru the messy integral -which eventually simplifed, and gave a total charge induced on the sphere = q.
When I saw this result I figured that if it simplified to this, it should have been obvious without going thru the integral. If I were to guess, I would have guessed that the total charge on the sphere should hve been q', not q. But even this isn't clear to me.
But anyway, was there a simpler way to show what the TOTAL charge on grounded conductor would be when a charge q sits some distance x away from it? I was thinking of Gauss' law, but that doesn't give me the answer I'm looking for.
Thanks
You can use the method of images for this, giving you an image charge q' located at a distance x' from the center of the sphere. The answer I get is:
q' = -qa/x
x' = a^2/x
(this doesn't really matter for my question)
Anyway, solving for the total charge induced on the sphere (by evaluating E at r=a, giving sigma, then integrating over the area of the sphere), it took pages and pages of grinding thru the messy integral -which eventually simplifed, and gave a total charge induced on the sphere = q.
When I saw this result I figured that if it simplified to this, it should have been obvious without going thru the integral. If I were to guess, I would have guessed that the total charge on the sphere should hve been q', not q. But even this isn't clear to me.
But anyway, was there a simpler way to show what the TOTAL charge on grounded conductor would be when a charge q sits some distance x away from it? I was thinking of Gauss' law, but that doesn't give me the answer I'm looking for.
Thanks