A proton strikes another proton at rest. Assume impact–parameter to be zero, i.e. head on collision. How close will be incident proton go to other proton?

When a proton collides head-on with another proton at rest, the interaction can be analyzed using concepts from classical physics and quantum mechanics. The key aspect to consider here is the nature of the forces at play, particularly the electromagnetic force due to the positive charge of both protons. Let's break this down step by step.

The Basics of Proton-Proton Interaction

Protons are positively charged particles, and when they approach each other, they experience a repulsive electromagnetic force. This force increases as they get closer together. In a head-on collision scenario, we can think of the incident proton approaching the stationary proton directly along a straight line.

Understanding the Concept of Closest Approach

The closest distance that the incident proton can reach to the stationary proton before being repelled is known as the "closest approach." This distance can be calculated using energy conservation principles and the concept of potential energy in an electric field.

Energy Conservation Principle

In this scenario, we can equate the kinetic energy of the incoming proton to the electric potential energy at the point of closest approach. The kinetic energy (KE) of the incident proton can be expressed as:

• KE = (1/2)mv²

where m is the mass of the proton and v is its velocity.

The electric potential energy (PE) between two protons at a distance r is given by:

• PE = k * (q₁ * q₂) / r

Here, k is Coulomb's constant, and q₁ and q₂ are the charges of the protons, which are equal to the elementary charge e.

Setting Up the Equation

At the point of closest approach, all the kinetic energy of the incident proton is converted into electric potential energy:

• (1/2)mv² = k * (e²) / r

Rearranging this equation allows us to solve for the distance r:

• r = (2k * e²) / mv²

Calculating the Closest Approach

To find the exact value of r, we need to plug in the values for the constants:

• Coulomb's constant, k ≈ 8.99 × 10⁹ N m²/C²
• Elementary charge, e ≈ 1.6 × 10⁻¹⁹ C
• Mass of a proton, m ≈ 1.67 × 10⁻²⁷ kg

Assuming the incident proton has a certain velocity v, you can substitute these values into the equation to find r. For example, if we assume a velocity of 1 × 10⁶ m/s, we can calculate:

• r = (2 * 8.99 × 10⁹ N m²/C² * (1.6 × 10⁻¹⁹ C)²) / (1.67 × 10⁻²⁷ kg * (1 × 10⁶ m/s)²)

After performing the calculations, you would find that the closest approach distance is on the order of femtometers (10⁻¹⁵ meters), which is typical for nuclear interactions.

Final Thoughts

This analysis shows how fundamental forces govern particle interactions at very small scales. The closest approach distance is a crucial concept in understanding nuclear physics and the behavior of subatomic particles. If you have any further questions or need clarification on any part of this process, feel free to ask!