A proton and an alpha particle are released from rest when they are 0.225 nm apart. The alpha pa

A proton and an alpha particle are released from rest when...

Key Concepts

Coulomb's Law

Coulomb's Law states that the electrostatic force between two point charges is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This principle is used to determine the force acting on charged particles, which is fundamental in calculating both accelerations and energy exchanges in systems of charged particles.

Conservation of Energy

The conservation of energy principle explains that as the charged particles repel or attract each other, the total energy of the system (the sum of kinetic and potential energies) remains constant. In problems involving electric forces, the initial electrostatic potential energy is gradually converted into kinetic energy as the particles accelerate. This principle is crucial for determining the maximum speed of the particles from the initial potential energy available.

Dynamics of Charged Particles

The dynamics of charged particles involves understanding how forces act on the particles and how their masses and charges determine their accelerations and motions. The acceleration at any point is given by Newton’s second law, where the force from the electric field is divided by the particle’s mass. This is particularly important when analyzing situations where particles have different masses and charges.

Dependence of Acceleration on Distance

In systems where forces obey an inverse-square law, such as electrostatic forces, acceleration is greatest when the particles are closest together. This means that the maximum acceleration occurs at the initial moment when the distance between the charges is minimal. As the particles move apart, the force and consequently the acceleration decrease.

Two-Body Problem in Electrostatics

The two-body problem in electrostatics involves analyzing the motion of both interacting particles simultaneously. In such problems, conservation laws (like momentum and energy) are used along with individual equations of motion to determine quantities such as maximum speed and acceleration. The different masses and charges affect how each particle responds to the electrostatic force in a way that respects both conservation of momentum and energy.

Transcript

00:01 Okay, so on this problem we have a proton of and a alpha particle distance 0 .225 nanometers apart.

00:20 We're told that the alpha particle has four times the mass and two times the charge as the proton.

00:29 So the proton has charge e, the n mass mass of the proton, and the alpha particle has charge 2e, and 4 times the mass of the proton.

00:49 And all of those are constants that we're going to look up later.

00:54 So we're told to calculate the maximum acceleration and velocity of each of the the particles after they're released from each other this distance apart.

01:09 So first thing that you should think of as a relevant equation is the force between two charged particles.

01:19 That is 1 over 4 pi epsilon not times the product of their charges divided by their distance squared.

01:39 Okay, so this force is going to decrease as these two positively charged particles move apart from each other.

01:48 So if we plot distance between them and force, they're going to start at 0 .225 nanometers.

01:58 They're going to have a certain force, and as they move apart, they're going to experience less and less force.

02:07 So because f equals ma, the force and the acceleration are just proportional to each other.

02:17 And so since the maximum force happens when they're first released, that's also when the maximum acceleration is going to happen.

02:26 So we know that the maximum acceleration is happening when they're 0 .225 nanometers apart.

02:32 So we just need to calculate the force and divide by mass to get the maximum acceleration.

02:41 So let's go on to another screen in order to plug in some numbers.

02:53 Okay, so we said that acceleration max is going to equal f max divided by the mass of whichever particle we will.

03:10 Looking at and from newton's third law we know that this max force is going to be the same for both particles but opposite end direction and we know that this f max is given by the equation we showed before here to be one over four pi epsilon not times the product of the charges let's put q proton and q alpha particle as variables for now divided by the distance between them squared, which we know now is just this .225 nanometers.

03:58 Okay, so we can look up some of these constants and plug this in.

04:02 So the charge of a proton is the same as the charge of an electron is 1 .2 .5 nanometers is 1 .5 .5.

04:16 0602 times 10 to the minus 19 coulamps.

04:23 We also need to know epsilon not, which is 8 .854 times 10 to the minus 12, kulam squared per newton meter squared.

04:49 And we're going to look for all of these in si units so that our life is easier.

04:55 The mass of the proton we're also going to need for this part is 1 .673 times 10 to the minus 27 kilograms.

05:15 Okay, so now we just need to plug in all of these numbers.

05:21 Let's see if i can move this to another screen.

05:28 Probably not.

05:29 Okay, so let's just let's just just plug them in from here.

05:36 We know that acceleration max of, let's do the proton first, is this f max divided by the mass of the proton, which is going to be the mass of the proton.

06:00 Let's put that out front is 1 .673.

06:11 Times 10 to the minus 27 kilograms.

06:17 I'm going to leave all the units out here, and maybe we'll look at them later.

06:22 And then we just have to write out this force part.

06:25 So 1 over 4 pi epsilon not, which is 8 .854 times 10 to the minus 12, at times the product of their charges, which is going to be the electron charge times two times the electron charge since we know the alpha particle is twice the charge of the proton, which is the same as two times the square of the electron charge.

07:03 To simplify that a little bit, 1 .602 times 10 to the minus 19.

07:14 Okay, now i'm getting squished in the corner.

07:16 But that's all squared because we have two times electron charge times one times electron charge.

07:25 And that's all divided by their distance squared, which we're going to put that in meters because we're working in si units.

07:32 0 .225 nanometers is 0 .225 times 10 to the minus 9 meters.

07:41 And that needs to be squared too.

07:43 Okay, so if we plug that into some sort of calculator, we can calculate that to be 5 .45 times 10 to the 18, and we're calculating an acceleration.

08:04 So that's going to be in units of meters per second squared since we're in si units.

08:13 So this is the acceleration of the proton.

08:16 We still need to calculate the acceleration of the alpha particle.

08:21 But luckily we know that these forces are equal, which will simplify our calculation.

08:29 So acceleration max of the alpha particle is going to be f max divided by mass of the alpha particle.

08:42 And now we could just plug this all in again, but to save us a little bit of work, let's just write this as f max of the alpha particle divided by the mass of the proton, because that's what we know.

08:56 That's this thing we just calculated, and then just multiply it by mass of proton divided by mass of the alpha particle.

09:04 So this equals this, just because we multiplied by a massive proton divided by a massive proton, which is one.

09:11 And we know this part is this answer we just calculated.

09:15 This part, we're told the alpha particle is four times the mass of the proton.

09:19 So this is just 1 divided by 4.

09:23 And so this answer is just this answer divided by 4, which is approximately 1 .36 times 10 to the 18 meters per second squared.

09:39 Okay.

09:40 And the question also asks where this maximum acceleration occurs.

09:45 And like we said before, the force is highest when they're closest together because of this one divided by r squared term.

09:54 And that is at this point 225 nanometers when they're first released.

10:02 Okay, so that's the acceleration part.

10:04 Now we just need to do the velocity part.

10:09 So since we know that this acceleration is going to be positive, no matter what r is, it's going to get smaller and smaller, but it's still going to be positive.

10:21 That means that both particles are going to be speeding up from the time they're released until they reach infinity.

10:30 So this maximum velocity is going to happen as far away apart as they can be because they're continually speeding up.

10:45 So that answer is the last part of the question already.

10:48 We know that the maximum velocity is going to be at infinity and not when they're released.

10:56 And we just need to calculate what that velocity is going to be.

11:00 So from conservation of energy, we know that the potential energy of this configuration is going to equal the sum of the kinetic energy of the particles when they're at infinity, and this is zero.

11:19 So let's write that out.

11:23 The kinetic energy of the proton plus the kinetic energy of the alpha particle equals this potential energy in the starting configuration.

11:38 Since when the particles reach infinity, they don't have any potential energy, all of their energy is going to be in the kinetic energy.

11:54 So what's the expression for this from your textbook? this is just 1 over 4 pi epsilon not times the product of the charges.

12:15 Divided by their distance.

12:17 So this should look very familiar to the force equation.

12:21 It's just divided by r instead of divided by r squared.

12:26 Okay.

12:27 So right now, we have all of these numbers except for the final velocity of the two particles.

12:35 And that's a problem because we only have one equation.

12:38 We can't solve for two unknowns.

12:40 So we need another equation...

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