Show all steps please The electric potential of two connecte

The electric potential of two connected conductors is Equal!! Consider two charged conducting spheres, radii r1 and r2, with charges q1 and q2. respectively. The spheres are far away from each other but connected with a very thin conducting wire. Knowing that the electric field of a charged sphere, outside the sphere is given by E(r)= where q is the total charge on the sphere and r the distance from the center, calculate the electric potential V(r1) and V(r2) just on the surface of each sphere as a function of (q1, r1) and (q2,r2), respectively. Choose a reference at infinity when calculating V from E, (V(infinity) = 0), and ignore the effect of the other sphere when doing the calculation for each. Knowing that the spheres arc connectcd with a conducting wire, what can you say about the voltage difference between them? Use parts a and b to calculate the ratio of q1 and q2 in terms of r1 and r2. Use parts a,b, and c to calculate the ratio of the electric fields just outside the surfaces of spheres, in terms of r1 and r2.

Solution

For any charged sphere with charge q and radius r, the electric potential at its surface is given as:

V = kq/r also the electric field at the surface is given as:

E = kq/r^2

Plus, it needs to be understood that for connected conductors with no source of charge, the potential difference between them happens to be zero. That happens once the charges redistribute themseleves on connection. Had the potential difference not been zero, we would have got a current flowing between them without a source of charge. We will use the above reasoning to solve the given problems as follows:

Part a.) Assuming that this part is to be solved without taking into consideration that the spheres are connected:

We have V(r1) = kQ1 / R1

Also, V(r2) = KQ2 / R2

Part b.) On the connection being made, both the spheres, as discussed above, need to be at the same potential.

That is, V(r1) = V(r2)

Part c.) Using parts a and b above, we have:

kQ1 / R1 = K Q2 / R2

or, Q1 / Q2 = R1 / R2 which is the required ratio

Part d.) Using parts a, b and c

we have the ratio of the electric fields as: E(r1) / E(r2) = (Q1* R2^2 / Q2* R1^2)

Using the ratio for Q1 / Q2, we get:

E(r1) / E(r2) = R2 / R1 which is the required ratio