Two very small 8.55- g spheres, 15.0 cm apart from center to center, are charged by adding equal

step 1 Hello, today we're going to be looking at a simple physics problem. So we have two charged spher

Two very small 8.55- g spheres, 15.0 cm apart from center to...

Key Concepts

Electrostatic Repulsion

When objects are charged with like charges, they repel each other due to the nature of the electrostatic force. Understanding that the spheres repel each other—and thus will accelerate away from one another—is crucial for determining the direction of acceleration in this problem.

Unit Conversion in Physical Calculations

Accurate problem solving in physics often requires converting given quantities into consistent systems of units (such as kilograms for mass and meters for distance). This ensures that calculations involving forces and accelerations using physical laws like Coulomb's and Newton's are correctly computed.

Electric Charge Quantization

Electric charge quantization is the principle that charge exists in discrete amounts, typically multiples of the elementary charge (the charge of an electron). This concept is important in problems where electrons are added to an object, as it directly connects the number of electrons to the total charge imparted on the sphere.

Newton's Second Law

Newton's second law provides the relationship between force, mass, and acceleration (F = ma). It allows us to calculate the necessary force to achieve a specified acceleration given the mass of an object. This concept is essential when linking the force from electrostatic interactions to the required acceleration of the spheres.

Coulomb's Law

Coulomb's law describes the electrostatic force between two charged objects. It relates the magnitude of the force to the product of the two charges and the inverse square of the distance between their centers. This law is key for determining the force produced when electrons are added to the spheres and is foundational to problems involving charged particles.

Transcript

00:02 Hello, today we're going to be looking at a simple physics problem.

00:07 So we have two charged spheres at 8 .55 grams each, and we're adding electrons to them at the same time until we can release them at 25gs.

00:26 The two particles are located at 15 centimeters apart from each other, and there are no initial forces or any other forces on the system.

00:40 So the question is how many protocols of electrons do we need to add to each of them for the outcome when released to be 25gs and what is the direction at which they'll be traveling? so we can answer that second question right away.

00:57 Because the charges that we'll be adding are the same, and we know that like charges repel each other, so when we release them, they will be shooting away from each other.

01:12 All right, so now let's take a look at how we can answer the first question.

01:18 So the first thing is recognizing that this problem involves two sets of laws, one is in electromagneticism, and one is in mechanics.

01:29 The two laws are kulam's law and the newton's second law.

01:33 And they can be described as this.

01:37 So kulam law states that force is equal to a constant times the product of the two particles charges divided by their distance squared.

01:52 And this constant here is just a constant of proportionality.

01:58 In fact, he has the name of kulam's constant.

02:03 And you can see that it is inversely proportioned to the distance.

02:09 So all of this combined together gives the force that's actually between the two particles.

02:17 And this is the same force that will drive this acceleration that we'll be looking at here.

02:23 Similarly, in newton's second law, newton's second law is what actually gives the relationship between mass and acceleration and force.

02:34 So we know that force is equal to mass times acceleration.

02:39 It's as simple as that.

02:41 All right, let's take a look at what we're given in this problem so that we can start filling out these equations.

02:49 All right, so we're given that the acceleration that we want is, 25 gs.

02:55 I will get into the units in a bit, but for now, just think about the physical quantities that we're dealing with.

03:04 The mass for each particle is 8 .55 gram.

03:08 In fact, the problem is symmetric, so it's sufficient to look at one particle, the properties of one particle is enough.

03:18 And the distance between the two particles for this r over here is 15 centimeter.

03:27 So moreover, there are two more constants that we'll have to deal with.

03:32 And they are first the kulums constant, and that is this k in kulom's law.

03:37 And the kulom's constants is just a number, a number of proportionality.

03:43 And it has the value of about 9 times 10 to the power of 9, newton meter square per column squared.

03:56 So you can write this as a product of inverse column squared, or you can write it as a in the reciprocal.

04:10 The second concept of goal needs, is the charge of a single electron here.

04:15 So the goal is to find out how many electrons we need to add.

04:20 So we all need to know what is the charge of single electron.

04:25 And lastly, to actually get the total charges, we just multiply the number of electrons by each individual electric charge.

04:44 All right, so let's take a look at how we can combine these two equations to get the result we want.

04:54 So firstly, we can write that, you can rewrite this top equation here.

05:00 So that's f equals to k, q1, q2 over r squared.

05:11 So we already know that q1 and q2 are the same.

05:15 So the problem is set up symmetrically.

05:17 So we're adding one electron at a time.

05:19 So we can rewrite that with the equation that we just introduced here.

05:24 Introduced here, which is n -e, n times e, number of electrons times the charge of electron over r squared.

05:43 So we can simplify this a little bit, so it's less cluttered, just n -e are altogether squared.

05:57 And all of this, imagine, this is the force that's actually acting on the particles.

06:04 So when we release it, the particles will accelerate under this force.

06:09 So that will make us equate this force to the force in newton's second law.

06:16 And that is simply m .a.

06:21 So now we have our entire equation with all of the known parameters as listed here.

06:28 And the only unknown that we're trying to solve for is this n.

06:34 All right, so we can rewrite this equation and solving for n.

06:40 So what is that? let's just go to a new page.

06:45 And that is n equals to, we bring the terms to the other side.

06:56 We know n has a squared, so we'll have a, you need to have a square root over the whole thing.

07:00 Square root of the mass times acceleration, times the radius squared, all over k, the equivalent constant, and the charge of an electron squared.

07:15 So because we have a square and inside a square root, so we can bring out those terms.

07:20 So that's r over e times the square root of m times a over k.

07:34 And that's it.

07:36 So basically, if you plug in all of the numbers on the previous page into this equation, actually, let me just write them down.

07:47 R is 15 centimeters.

07:55 The charge of electron is 1 .6 times 10 to the power of minus 19.

08:03 It's because an electron is a very small thing.

08:07 Coolums all inside the square root.

08:14 I'm going to close it later.

08:16 So each thing is 8 .5 grams and we have the acceleration that we're looking for.

08:28 That's 25 gs.

08:32 So it's very confusing when we have the two gs over here, but i'll explain in a little bit of how we can get rid of those.

08:40 And lastly, and we want to divide the whole thing by the charge.

08:46 The the coolum's constant, which is 9 times 10 to the power of 9 newton meter squared per coulum squared.

09:04 And that's it.

09:08 So after all the units cancel out, we all get our n.

09:13 So now you might be asking, how do we actually do cancel the units? the easiest way is to actually convert everything to the same units.

09:27 So let's take, we'll do metric for example.

09:30 So for instance, this cm here, so we're going to cut out a little bit, this cm, we know that this, we want this unit to be the same as this unit, this meter unit here.

09:49 So how do we do that? let's deal with unit conversion.

09:57 So what we want to do for the sake of simplicity is to convert one unit to another, for example.

10:09 A very easy way of doing that is to recognize that when you multiply the entire equation, multiply everything by one, you will not change the result.

10:23 And let's take a look at what sort of quantities we can convert between units, in order to have the solution equals to 1.

10:34 So we have, let's say we want to convert from cm to meters.

10:41 So we write down one cm, so we can try to solve for this equation here.

10:51 So we know that multiplying the whole thing by the quantity 1, will not change the equation.

10:55 So what quantity gives the same cm as meters? meters.

11:02 So and that is we have 100 cm per meter.

11:06 So that's an easy way of writing it per one meter.

11:12 So we can have this.

11:14 So multiplying the whole equation by this will not change the thing.

11:18 So in fact, in this case, we actually want the reciprocal of that.

11:23 So we want meters to be on top rather than on the bottom.

11:30 So let's move this to the bottom.