Electron Force and Electric Field

[Paul2011]

Homework Statement

A) Three conducting spheres, a, B , and c, form
an equilateral triangle of side length "l" and have
initial charges of -2q, -4q, and 8q, respectively.
Show that the magnitude of the electrostatic force
between spheres  a and c is given by

F_ac= (4q^2)/(π€_o l^2 )

B) The following steps are then taken:  a and b are
connected by a thin wire and then disconnected;
b is grounded by the wire and the wire is then
removed; b and c are connected by the wire and
then disconnected. Show that the magnitudes of
the electrostatic forces between spheres a and c
and between spheres b and c, respectively, are
given by

F_ac=(3q^2)/(π€_o l^2 )
F_bc=(4q^2)/(π€_o l^2 )

Homework Equations

I think I got part A right, but part B i seem to be stuck unfortunately.
Any help would be appreciated, thanks.

The Attempt at a Solution

A)

F_ac=(-2q)(8q)/(4π€_o l^2)
F_ac=(4q^2)/(π€_o l^2)

 

[soothsayer]

So, in part B, what I'm assuming is happening is that as the spheres are being connected and disconnected, charge is being transferred so that the charge of both of the spheres that are connected are the same. So as A and B are connected, their charges both become -3q. As B is grounded, it loses all its charge, but then is connected to C, which evens out so that B and C have a charge of 4q. The end result is that the charge on A is -3q, the charge on B is 4q and the charge on C is 4q.

Now do what you did last problem for the force and you'll have your answers!

 

[Paul2011]

Ok, so this is what I've done based on your insight (thanks btw).

(a+b)/2 = (-2q-4q)/2 = -3q

a=-3q , b=-3q, c=8q

a=-3q, b=0q, c=8q

(c+b)/2 = (8q+0q)/2 = 4q

a=-3q , b=4q, c=4q

F_ac = (-3q)(4q)/4π€_o l^2
F_ac = 3q^2/π€_o l^2

F_bc = (4q)(4q)/4π€_o l^2
F_bc = 4q^2/π€_o l^2

looks right to me, thanks again. I hope its right at least..

 

[soothsayer]

Yep, you got it. No problem.

 

[rwisz]

B)

To solve for the electrostatic forces between spheres a and c, and between spheres b and c, we need to consider the changes in electric charge and the resulting electric fields.

1. When a and b are connected by a thin wire, the charges on the two spheres will equalize and become -3q and -3q, respectively. This is because the charges on the wire will redistribute themselves to balance out the charge on the spheres.

2. When b is grounded and the wire is removed, the charge on b will be neutralized, leaving only the charge of -3q on sphere a.

3. When b and c are connected by the wire, the charges on the two spheres will equalize again and become -3q and 11q, respectively.

Using Coulomb's Law, the magnitude of the electrostatic force between spheres a and c can be calculated as:

F_ac=(-3q)(11q)/(4π€_o l^2)
F_ac=(33q^2)/(4π€_o l^2)

Similarly, the magnitude of the electrostatic force between spheres b and c can be calculated as:

F_bc=(-3q)(11q)/(4π€_o l^2)
F_bc=(33q^2)/(4π€_o l^2)

Therefore, we can see that the magnitudes of the electrostatic forces between spheres a and c and between spheres b and c are both given by (33q^2)/(4π€_o l^2), which simplifies to (3q^2)/(π€_o l^2).