A 4.78-MeV alpha particle from a 226 Ra decay makes a head-on collision with a uranium nucleus.

step 1 In this problem, we are asked to find the distance of the closest approach of the alpha particle

A 4.78-MeV alpha particle from a 226 Ra decay makes a...

A $4.78-\mathrm{MeV}$ alpha particle from a 226 $\mathrm{Ra}$ decay makes a head-on collision with a uranium nucleus. A uranium nucleus has 92 protons. (a) What is the distance of closest approach of the alpha particle to the center of the nucleus? Assume that the uramium nucleus remains at rest and that the distance of closest approach is much greater than the radius of the uranium nucleus. (b) What is the force on the alpha particle at the instant when it is at the distance of closest approach?

A $4.78-\mathrm{MeV}$ alpha particle from a 226 $\mathrm{Ra}$ decay makes a head-on collision with a uranium nucleus. A uranium nucleus has 92 protons. (a) What is the distance of closest approach of the alpha particle to the center of the nucleus? Assume that the uramium nucleus remains at rest and that the distance of closest approach is much greater than the radius of the uranium nucleus.
(b) What is the force on the alpha particle at the instant when it is at the distance of closest approach?

Key Concepts

Distance of Closest Approach

The distance of closest approach is the minimum separation between a moving charged particle and a target charge during a collision. It can be determined by equating the particle's initial kinetic energy with the electrostatic potential energy at that point. This concept is critical in scattering experiments as it provides insight into the range and strength of the interaction between the charged entities.

Electric Force Calculation in Collisions

Calculating the electric force during collisions involves using Coulomb's law at the point where the interacting particles are at their closest approach. Knowing this force allows for the understanding of the instantaneous interaction dynamics and can help predict the subsequent motion of the particles after the collision.

Coulomb's Law

Coulomb's law provides the mathematical relationship between the electric force and the charges involved, stating that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is fundamental for calculating both the force during the interaction and the related potential energy, which is key to solving for parameters like the distance of closest approach.

Electrostatic Potential Energy

Electrostatic potential energy is the energy stored between charged particles due to their positions relative to each other. For two point charges, this potential energy is given by an expression involving the product of the charges and inversely proportional to their separation distance. In collision problems, it is used to determine how the particle’s kinetic energy transforms as it slows down due to repulsive forces.

Energy Conservation

Energy conservation is the principle stating that the total energy in an isolated system remains constant. In scattering problems, this means that the kinetic energy of an incoming particle can be completely converted into another form of energy—typically potential energy—when it encounters a force field, such as the electrostatic field in a collision between charged particles.

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