Copper atoms have 29 protons in their nuclei. If the copper...

Copper atoms have 29 protons in their nuclei. If the copper nucleus is a sphere with a diameter of $4.8 \times 10^{-15} \mathrm{m},$ find the work required to bring an alpha particle (charge $=+2 e )$ from rest at infinity to the "surface" of the nucleus.

Key Concepts

Approximation of Spherical Charge Distribution as a Point Charge

In many electrostatic problems involving spheres, especially when evaluating interactions at or outside the surface, the charged sphere can be approximated as a point charge located at its center. This simplification is based on the symmetry of the sphere and greatly simplifies the calculations, as it allows the use of formulas derived for point charges to estimate the electric potential and energy interactions.

Work in Electrostatics

The concept of work in electrostatics involves the energy required to move a charge against the electric field. When a charge is moved from infinity (where the potential energy is defined to be zero) to a point closer to another charged body, the work done is equal to the increase in electric potential energy of the system. This principle underlies the calculation of the energy necessary to bring an alpha particle to the surface of a nucleus.

Electric Potential Energy

Electric potential energy is the energy a charged particle possesses due to its position in an electric field created by another charge. In electrostatics, this energy determines the work required to move a charge from one point to another, such as from infinity to the surface of a charged sphere. The potential energy is calculated using the charge values and the distance between them, following the inverse proportionality of the distance.

Coulomb's Law

Coulomb's Law describes the force between two charged particles. It states that the electrostatic force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This law is fundamental in calculating the interactions between charges in any electrostatic problem, including the scenario of bringing an alpha particle close to a charged nucleus.

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