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So in our physics class today, we had a demonstration involving a charged rod and a neutral ball attached to a string (a physical pendulum). At first, when the rod was placed near the ball, the ball was attracted to the rod until the two objects "touched" and the neutral ball gained the same charge as the rod and was consequently repelled.

It is my understanding that from the perspective of atoms and electrons the objects never actually "touch." So something must happen at certain distance where electrons go from just repelling each other to actually being transferred.

So here are my questions: What exactly defines this threshold where charge is transferred between two objects? What forces and principles are at play that determine this? And finally, do we know any mathematical equations we know that can model or govern this?

Any help is appreciated, and if you could tailor your response to the understanding of someone who is about mid-way through high school AP Physics, I would be grateful.

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The micro world of atoms and molecules is ruled by quantum mechanics. The forces controlling the interactions between them at the level affecting everyday situations, as touching, are electromagnetic.

In quantum mechanics the electrons are in bound states around the atoms, and at most can be mobile in bands when in a solid, i.e a bound state due to the effective potentials of the lattice of the solid which seems continuous .

So here are my questions:

What exactly defines this threshold where charge is transferred between two objects?

Electrons to move from one solid object to another when brought into a close contact have to find free energy levels that they can occupy. Otherwise they will not transfer, as happens with insulators. As the rod with the charge touches the pendulum electromagnetic interactions determine the behavior of the extra electrons in the rod: the threshold is that the electrons of the rod need enough energy to match the energy needed to join a band in the pendulum.

You do not state the material of the rod and the pendulum. Even though an object is neutral the forces around atoms and molecules are not symmetric, and collective fields of positive charge can appear to be attracted by the negative one in the ball.

What forces and principles are at play that determine this?

explained above.

And finally, do we know any mathematical equations we know that can model or govern this?

Yes , it is the quantum mechanics band theory of solids that is used for such situations, which is well studied and validated.

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