An alpha particle with a kinetic energy of 10.0 MeV makes a...

An alpha particle with a kinetic energy of 10.0 MeV makes a head-on collision with a gold nucleus at rest. What is the distance of closest approach of the two particles? (Assume that the gold nucleus remains stationary and that it may be treated as a point charge. The atomic number of gold is $79,$ and an alpha particle is a helium nucleus consisting of two protons and two neutrons.)

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. In the context of scattering problems, the kinetic energy of an incoming particle determines how close it can come to another particle before being repelled, as it is converted into potential energy during the interaction.

Coulomb Potential Energy

Coulomb potential energy is the energy stored due to the electrostatic interaction between charged objects. It is calculated using Coulomb's law and plays a critical role in collisions between charged particles, where the repulsive force between like charges converts the particle's kinetic energy into potential energy.

Energy Conservation in Scattering

Energy conservation is a fundamental principle asserting that the total energy in an isolated system remains constant. In the context of charged particle collisions, the initial kinetic energy of the projectile is entirely transformed into Coulomb potential energy at the point of closest approach, allowing for the determination of that distance.

Distance of Closest Approach

The distance of closest approach is defined as the minimum separation between two interacting particles during a collision. It can be calculated by setting the initial kinetic energy equal to the Coulomb potential energy at that point, thereby providing insights into the interplay between the repulsive forces and the particle’s initial energy.

Transcript

00:01 In this question, we have a collision between an alpha particle.

00:04 So a alpha particle can be treated just as a helium nucleus.

00:10 And then we also have, what we would have is a gold nucleus.

00:16 So gold, i think, is a u.

00:23 So they're just a nuclear.

00:25 So it's not an atom.

00:27 A nucleus means it only has positive charges.

00:30 So the helium nucleus will carry plus 2e and the gold will have plus 79e.

00:39 So what happens is that they initially has a kinetic energy of 10 .0 m .ev.

00:47 But then they collide.

00:49 And when they collide, you do to know that because they are both positive charges, they will repel on each other.

00:57 So when one comes in the same direction, they're the other one comes in the same direction.

01:01 So they're not going to actually like completely touch each other because the closer they get, the larger the electric force is going to be.

01:10 And when they're infinitely small, or when they're infinitely close, the electric force between them is like infinity and that's not possible.

01:18 So they will be sufficiently close when the electric field, when the electric force between them, the repulsion becomes too close.

01:27 They will turn and go back.

01:33 So this question wants us to find at which point will they start to turn? so what is the closest approach? and that happens when the electric potential energy, ue, which is kq1, q2 over r, is exactly the same as the kinetic energy.

01:54 So from this expression, we can find the r, r equals k k k k u2 over k and q1 q2 over k and q1 q2 what is q1 q2 q1 q2 is 1 is 2 e 1 is 79e divided by k okay okay uh this is a small k just the constant we're used to this is large k meaning the kinetic energy okay so a small k is 90 times 10 to the nice and 2 times 79 times e is 1 .6 times 10 to the negative 19 square over big k is 1 .60 times 10 to the negative uh big k is 10 to the negative 12 yeah so um okay let's say um okay let's maybe the best way is do it...

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