Imagine a sphere or radius R filled with negative charge of uniform charge densi
ID: 1894857 • Letter: I
Question
Imagine a sphere or radius R filled with negative charge of uniform charge density. The total charge is equivalent to the charge of two electrons. Now insert two protons into the sphere and assume that in spite of their presence, the negative charge distribution remains uniform. Where do you need to locate the protons so that the force on each of them is zero?
[Hint: The sphere is basically uniformly charged with total charge -2q, but the two protons that magically (magically because the charge distribution doesnt change) appear will repel each other. however, the negative charge density will push them together. there is a point where these two effects balance out (where the net force on each of the protons is zero). so that's how the problem is set up, but I have no idea how to proceed!]
Explanation / Answer
From simple symmetry we can think that charges must be diametrically opposite due to force of repulsion of protons.
Let, they are diametrically opposite on a sphere of radius r inside the sphere of radius R.
now, charge density= -2e/(4**R3/3)
chrge inside the sphere of radius r= -2e/(4**R3/3) * 4**r3/3 = -2e*r3/R3
Net charge inside the sphere= +2e- charge inside the sphere of radius r.
so, net charge= 2e -2e*r3/R3 = 2e(1-(r/R)3)
Since the negative charge density is uniform, so Electric field=zero.
so, net charge = 0.
r=R.
so, charges are on the surface of sphere at diametrically opposite ends.