(a) Use energy methods to calculate the distance of closest approach for a head-on collision bet

(a) Use energy methods to calculate the distance of closest...

Key Concepts

Closest Approach in Coulomb Collisions

The distance of closest approach in a Coulomb collision is the point where the kinetic energy of the incoming particle is completely converted into electrostatic potential energy. This concept is fundamental in calculating how close two charged particles can come to each other during a head-on collision, providing insights into the dynamics of nuclear interactions.

Coulomb Barrier

The Coulomb barrier refers to the repulsive energy barrier experienced by two positively charged nuclei due to their electrostatic interaction. Overcoming or reaching this barrier is critical in nuclear reactions and scattering experiments, as it determines the minimum energy or speed necessary for a particle to approach another nucleus closely.

Electrostatic Potential Energy

Electrostatic potential energy arises from the Coulomb interaction between charged particles and is given by a formula involving the product of the charges divided by the distance between them. This concept is essential for analyzing the energy conversion from kinetic to potential energy during the collision of charged particles.

Conservation of Energy

In nuclear collision problems, conservation of energy is used to equate the initial kinetic energy of the moving particle with the electrostatic potential energy at the point of closest approach. This principle allows one to determine how far a particle will travel into the repulsive field of another charged particle before coming to a stop momentarily.

Transcript

00:01 Okay, so to do this problem, a bit complicated, i'll do a six -step solution.

00:09 So for part a, we'll do step one.

00:14 What is the question asking for? so here we see it's looking for the closest distance for our head -on approach.

00:24 You need to know that refers to the electric potential.

00:28 So we're looking for that distance.

00:36 Remember, just pull up the variable that the question is asking you for.

00:39 And in part a, that is that distance r.

00:43 So step two, i will conceptualize the problem, and i'll use a drawing.

00:51 So this alpha particle is moving towards a gold particle, which is at rest.

01:05 And then afterwards, they have that electro -potential distance r.

01:16 Now step three, what variables does the problem give us? here we have the energy that the particle started with, which is kinetic energy since it was moving.

01:44 So let's call that kinetic energy subscript alpha equals 0 .5 mega electron volts.

02:07 Yep, and that's all we have so far to go off of.

02:12 Step four, now we are going to apply equations.

02:21 So again, looking from our conceptualized drawing here, we know that it went from kinetic in the alpha particle to electro potential.

02:36 So let's assume all the energy is conserved.

02:42 And we will say that kinetic energy in the alpha particle is equal to the electropotential.

02:57 So electric potential, the equation for that is coulum's constant times charge 1 times charge 2.

03:16 That's the charge of the two particles.

03:19 In this case, the alpha particle and the gold particle over the distance between them.

03:30 And this will be equal to the kinetic energy of the alpha particle.

03:40 So here we want one equation and one unknown.

03:43 So since we know we're solving for this r, i'm just going to rearrange this equation.

03:50 So r will equal.

03:54 We have the same variables in the numerator.

04:01 And then this kinetic energy comes down to the bottom here.

04:09 Okay, so one equation, one unknown, we have this kinetic energy value.

04:18 Kulums constant, we know to be 8 .99 times 10 to the 9 new ints times meter squared over kulam squared.

04:45 Okay, so now there's q1 and q2.

04:48 So let's say our q1 is the charge of the alpha particle.

04:57 So this would be the charge of the total numbers of protons.

05:02 So we would need to know the atomic number or number of protons in the alpha particle times the charge of a proton.

05:26 So we know that an alpha particle has two protons.

05:29 And the charge of a proton is 1 .6 times 10 to the negative 19 coulums.

05:47 Again, these are things you can look up in your physics book, find online in a quick search.

05:53 So now we know this value q1.

05:59 Now q2 will be the charge of our gold particle, which would be the atomic number of gold times the charge of a product.

06:23 Proton.

06:26 So we know that gold has 79, 79 protons and the same charge 1 .6 times 10 to the negative 19.

06:54 So then we can say that q1 times q2 is equal to both of these atomic numbers alpha times gold times the charge squared if you rearrange these two equations for q1 and q2 and which we all know all these values here.

07:30 We know the atomic number for alpha particle and of gold, and we know the charge of a proton.

07:40 So now we have one equation, one unknown, and we are ready to solve in step five.

07:59 But before i do that, before you are ready to solve, you want to make sure you do a quick check of your units.

08:05 So as you can see here in the kulums constant, we have newtonmeter squared over kulam squared, which we know will probably cancel out down here.

08:20 But for our kinetic energy, we have that in mega electron volts.

08:29 So we might want to convert that into joules and then break that down again into newton meters.

08:42 So if we multiply this, do a conversion factor right here.

09:16 So in one electron volt, there's 1 .6 times 10 to the negative 19 joules.

09:20 So we're just converting here to get it into jewels.

09:28 And we also know that one jewel is equal to 1 newtometer.

09:38 So if we multiply 0 .5 times 1 .6 times 10 to the negative 19, we will get 8 times 10 to the negative 14 new meters.

10:03 Okay, so now we have everything that we need.

10:06 Everything is known on the right side of the equation and they're in the units that we want.

10:13 So now i'm going to plug this, plug everything in down here.

10:26 So r will equal 8 .99 times 10 to the 9.

10:42 And for the units, i'm going to do them in green.

10:50 Noometer squared over coulum squared.

11:01 And so b times q1 times q2.

11:09 That is the atomic number of both the alpha particle and gold.

11:17 So i'm going to replace that with times 2, times 79, and times a charge of a proton squared.

11:31 1 .6 times 10 to the negative 19, coulums squared.

11:46 Let's change its coulum to green, the units.

12:09 All right, i'm going to put all this over our kinetic energy from the alpha particle, which we found to be 8 times 10 to the negative 14 new meters.

12:31 Change this to green.

12:45 Ok, so for the units, we can see what cancels out here.

12:50 These newtons will cancel out.

12:54 This will make a coulum squared.

12:56 So this will cancel out with a coolum square down here...

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