the question couldn't fit so it's down below

Question

The dome of an electrostatic generator is spherical conductor of radius 13cm. It is initially charged so that the electric potential at the surface is 4.5kV. A smaller isolated sphere of radius 5.2 cm, initially uncharged, is brought near to the dome. Sparking causes a current between the two spheres until they reach the same potential. Assume that any charge on a sphere may be considered to act as a point charge at its centre. Calculate the charge that is transferred between the two spheres.

pls help, thank you in advance!

Answer 1

Let

R = 13 cm = 0.13 m be the radius of the larger sphere,

r = 5.2 cm = 0.052 m be the radius of the smaller sphere,

Q (unknown) be the initial charge on the larger sphere,

q (to be calculated) be the charge transferred to the smaller sphere,

U = 4.5 kV = 4500 V be the initial electrical potential on the larger sphere,

k = 9×10⁹ Nm²/C² be Coulomb's constant

To solve this problem you need two identities:

At a distance d from a point charge c

the electrical field has magnitude kc/d², and

the electrical potential is kc/d.

The latter allows us to calculate Q, because it gives us the equation

U = kQ/R

From that

Q = UR/k

To calculate q we use the condition that it will reach a value causing any current to stop.

At that point the smaller sphere will have charge q and the larger sphere Q-q.

Intentionally or unintentionally the condition for the current to stop is stated in a misleading way.

It says that it will stop when the two sphere will reach the same potential.

This may give to the temptation to express it by the condition

k(Q-q)/R = kq/r which is wrong!

Although the potential on the smaller sphere is kq/r, that is true only

in the absence of the larger sphere.

You cannot add up potentials due to different sources to get a net potential.

In contrast, you can add up forces or fields due to different sources,

because those are vectors, and you apply vector addition.

(In contrast, potentials are scalars which lost information about direction of forces.)

In general, electrical current stops when the net electrical field is 0.

(Which coincides with the two net potentials being equal.)

In our case the fields due to the two spheres have the opposite direction,

therefore they will add up to 0 when they have the same magnitude.

So we get the equation

k(Q-q)/R² = kq/r²

From that

q = Qr²/(R² + r²)

After substituting Q from above

q = URr²/k(R² + r²)

Substituting actual numbers

q = 4500×0.13×0.052²/(9×10⁹×(0.13² + 0.052²)) ≈ 9×10^-9 C