Electrostatics > The potential of an electric field. Conductors in a constant electric field > Problem 6.3.37

Condition

$6.3.37$. How, having a metal ball with charge $Q$, can one charge another conductor with a charge greater than $Q$?

Solution

Obviously, this cannot be done using only these two conductors – we would only have the option of touching the first to the second, and a smaller charge would flow onto it than required. Therefore, we add to the system an infinite reservoir at zero potential – the Earth – and another reservoir for intermediate storage of charge. Now, using the field of the charged ball, we can gradually transfer charge from the Earth to the reservoir, and then charge the second ball.

It is convenient to use a hollow metal sphere as a reservoir – if a charged body touches it from the inside, almost all the charge is transferred to the reservoir. This happens because inside a conductor in equilibrium the electrostatic field is zero, and excess charge is always pushed to the outer surface.

One possible method:

• Bring the charged (say, positively) ball close to an uncharged earthed conductor $A$, without touching it. Under the action of the ball's field, a negative charge $-q$ ($q < Q$) is induced in conductor $A$. That is, electrons flow from the Earth onto $A$ under the action of the ball's field.

• Break the connection with the Earth, and then remove the original ball. $A$ remains negatively charged.

• Transfer this charge $-q$ to the sphere by internal contact.

• The cycle can be repeated until the charge on the sphere becomes sufficiently large in magnitude.

After that, there are two options. If Savchenko meant a larger charge in magnitude, one can charge $A$ by simply touching it to the reservoir. If we need a charge of the same sign as $Q$, then we can earth $A$, induce the desired charge on it by bringing the sphere close to it, then break the connection with the Earth and remove the reservoir.