Adding a proton to a nucleus The nucleus of an atom is...

Adding a proton to a nucleus The nucleus of an atom is positively charged because it consists of positively charged protons and uncharged neutrons. To bring a free proton toward a nucleus, a repulsive force $F(r)=k q Q / r^{2}$ must be overcome, where $q=1.6 \times 10^{-19} \mathrm{C}$ is the charge on the proton, $k=9 \times 10^{9} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}, Q$ is the charge on the nucleus, and$r$ is the distance between the center of the nucleus and the proton. Find the work required to bring a free proton (assumed to be a point mass) from a large distance $(r \rightarrow \infty)$ to the edge of a nucleus that has a charge $Q=50 q$ and a radius of $6 \times 10^{-11} \mathrm{m}$

Key Concepts

Coulomb's Law

Coulomb's Law describes the electric force between two point charges, indicating that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This principle is fundamental in understanding how charged particles interact and is essential for analyzing the forces involved when bringing a charged particle close to another charged entity.

Electric Potential Energy

Electric potential energy is the energy stored in a system of charges due to their positions relative to each other. It quantifies the work required to assemble a configuration of charges by moving them from a reference point (commonly at infinity) to specific positions against the electrostatic forces present.

Work in Electrostatic Fields

The concept of work in electrostatic fields involves calculating the amount of energy needed to move a charge within an electric field against the electrostatic force present. This is typically accomplished by integrating the force over the path of the charge, which in many cases leads to evaluating the change in electric potential energy between two points.

Integration of Inverse-Square Law Forces

Integrating a force that follows an inverse-square law, such as the Coulomb force, is essential when calculating work or potential energy differences. This integration accounts for the varying magnitude of the force as the distance between charges changes, ultimately providing the total energy required to move a charge between two points in a radially symmetric field.

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