Two identical metal spheres have charges of q1 and q2. They...

Two identical metal spheres have charges of $q_{1}$ and $q_{2}$. They are brought together so they touch, and then they are separated. (a) How is the net charge on the two spheres before they touch related to the net charge after they touch? (b) After they touch and are separated, is the charge on each sphere the same? Why? Four identical metal spheres have charges of $q_{\mathrm{A}}=-8.0 \mu \mathrm{C}, q_{\mathrm{B}}=-2.0 \mu \mathrm{C}$ $q_{\mathrm{C}}=+5.0 \mu \mathrm{C},$ and $q_{\mathrm{D}}=+12.0 \mu \mathrm{C} .$ (a) Two of the spheres are brought together so they touch and then they are separated. Which spheres are they, if the final charge on each of the two is $+5.0 \mu \mathrm{C} ?$ (b) In a similar manner, which three spheres are brought together and then separated, if the final charge on each of the three is $+3.0 \mu \mathrm{C}$ (c) How many electrons would have to be added to one of the spheres in part (b) to make it electrically neutral?

Two identical metal spheres have charges of $q_{1}$ and $q_{2}$. They are brought together so they touch, and then they are separated. (a) How is the net charge on the two spheres before they touch related to the net charge after they touch? (b) After they touch and are separated, is the charge on each sphere the same? Why?

Four identical metal spheres have charges of $q_{\mathrm{A}}=-8.0 \mu \mathrm{C}, q_{\mathrm{B}}=-2.0 \mu \mathrm{C}$ $q_{\mathrm{C}}=+5.0 \mu \mathrm{C},$ and $q_{\mathrm{D}}=+12.0 \mu \mathrm{C} .$ (a) Two of the spheres are brought together so they touch and then they are separated. Which spheres are they, if the final charge on each of the two is $+5.0 \mu \mathrm{C} ?$ (b) In a similar manner, which three spheres are brought together and then separated, if the final charge on each of the three is $+3.0 \mu \mathrm{C}$ (c) How many electrons would have to be added to one of the spheres in part (b) to make it electrically neutral?

Key Concepts

Conservation of Charge

This principle states that the total electrical charge in an isolated system remains constant, no matter how the charge is redistributed among its components. In problems involving the contact of conductors, the sum of the charges before contact is equal to the sum of the charges after contact, regardless of how the charges are divided between the conductors.

Charge Redistribution on Contact

When two conductors are brought into contact, the free electrons move between them until they reach an equilibrium state. For conductors of the same size and shape, this generally results in an equal sharing of the total charge. This concept is crucial in understanding how charges distribute themselves in systems comprising identical conductors.

Electrostatic Equilibrium and Equalization of Potential

Upon contact, conductors rearrange their charges to reach an equal electric potential across their surfaces. When identical spheres are used, the condition of equal potential results in an equal division of the net charge. This is a direct consequence of the conductors having the same capacitance, leading to identical voltage when the charge is equally distributed.

Quantization of Charge and Electron Charge

Charge is quantized, meaning that it exists in discrete amounts typically measured in multiples of the elementary charge (the charge of a single electron). Understanding this quantization is important when converting between the macroscopic charge quantities (such as coulombs or microcoulombs) and the number of electrons transferred or added during a process, as it emphasizes that charges move in uniform, smallest possible amounts.

Transcript

00:01 In this problem, we're asked what happens when two identical metal spheres touch, and actually also with more than two touch.

00:11 First, we're asked about the net charge.

00:13 Net charge is just very simply when we just have two objects, q1 plus q2.

00:18 And we're asked what happens to that value when the two charges touch? and the answer is nothing.

00:26 Charge is conserved.

00:30 It does not, if there's no way for it to get away from it.

00:32 From those two objects, then whatever is there is always going to be there.

00:39 Q is conserved, constant, just like momentum is conserved, charge is conserved.

00:51 The next question asks, what happens to the actual individual cues? i'll call q1 and q2 prime, put primes on those.

01:00 Well, it turns out very simply when they're identical, and i'll take you through a simple situation in a moment.

01:07 Q1 prime, which is after the touching, is equal to q2 prime, is equal to half of the net.

01:14 Each one has half.

01:16 There's a sherry.

01:25 And let's try to make this plausible.

01:27 With any type of motion, not going to go through the actual, what happens in terms of how the charge is set up for the force, it's just to understand that there is such a thing.

01:37 When there's motion, you have to have, when there wasn't motion before, you have to have force, net force.

01:45 So let's just look at a simple example.

01:47 Say i have minus eight microculeomes on the left and i have plus five microculeomes on the right.

01:54 Just before they touch, this gives me q2 and this gives me q1.

01:59 So if i were to use my little formula, minus 8 plus 5 is minus 3 divided by 2 is minus 1 .5.

02:06 And that's what we should get in the end.

02:09 Okay.

02:11 Now let's look at what happens to a net charge a negative charge in the interface, whatever force is it feel.

02:19 Well, from the charges on its left, it's going to feel repulsive force to the right.

02:23 From the charges on its right, it's going to feel a tractor force to the right.

02:28 So net is to the right.

02:29 So negative charges are going to move from left to right.

02:33 So sometimes later, we're going to pick up the scene where maybe minus five microculeomes and move from left to right.

02:41 So this leaves me with minus three still.

02:44 You know, excess negative charge on the left, and minus five have moved from left to right, so that means i have no excess on the right.

02:55 It's a zero.

02:56 It doesn't mean there are charges.

02:57 Everything's just balanced.

03:00 Every plus is a minus and so on.

03:03 Now, is this the end? because i have a neutral material? no, because the negative charge sitting here will be feeling a repulsive force to the right still.

03:14 So negative charge is still going to be moving from left to right.

03:22 So as each negative charge moves from the left to right, i'm going to be getting a lesser negative, minus 3 to minus 2 .7, however you want to think about it.

03:32 And as those negatives move to the form of the neutral area, i'm picking up negative.

03:38 So if minus 1 moved, so i'm down to minus 2, and i got minus 1 on the right...

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