# Comments on the Magnitude of Induced Charge

Is it necessary that the induced charge be always less than or equal to the inducing charge? Why can't it be greater?

Could someone please explain why this happens, and to what extent can we comment on the magnitude of induced charge? If the shape and geometry of the body on which charge is being induced is specified, can we say more about how much charge exactly is induced?

It may seem obvious at the first glance, but I'm looking for a proper explanation - maybe mathematically rigourous if possible. I'm not able to find a proper, detailed explanation online, or in texts. In several places, just the statement is mentioned - without an explanation.

Thanks a lot.

## 1 Answer

First, let's clarify that a charge can be induced into a neutral conductor, only if this conductor is grounded. Otherwise the conductor will remain neutral.

Now, I'd like to propose an intuitive explanation based on the rules associated with electrical field lines.

Let's start with a charged conductive sphere in space. Its field lines will start at the surface of the sphere and will go to infinity. If the sphere is brought close to ground, its charges will shift to the side close to the ground and attract opposite charges to the surface of the ground.

I think we can safely assume that all the field lines leaving the sphere will end up on the surface of the ground and, moreover (and this is not exactly one of the well known rules) each positive unit charge on the sphere will be connected with the opposite unit charge on the surface of the ground. Let's call it a one-to-one rule.

Now, if we bring a grounded conductive object close to the charged sphere, some charge will be induced into that object, so at least some of the field lines leaving the sphere will jump from the ground to that object.

Now, since the charge on the sphere did not change and, therefore, the total number of field lines leaving the sphere did not change, we'll have to conclude that the number of lines ending up on the object won't be greater than that total number and, therefore (if we believe the one-to-one rule), the charge induced into the object won't exceed the charge on the sphere.

If another conductive sphere is brought in, the charges in the first sphere would shift on the side of the sphere closer to the second sphere and, I propose, all its field lines end up on the negative charges of the second sphere.