The nuclei of large atoms, such as uranium, with $92$ protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately $7.4\times10^{-15}$ m. What magnitude of electric field does it produce at the distance of the electrons, which is about $1.0\times {10}^{-10}$ m?

Verified step by step guidance

1. First, recognize that the electric field produced by a spherically symmetric charge distribution outside the sphere can be treated as if all the charge were concentrated at the center. This is due to Gauss's Law.
2. Calculate the total charge of the uranium nucleus. Since uranium has 92 protons, and each proton has a charge of approximately 1.6×10−19 C, the total charge Q is given by Q = 92 × 1.6×10−19 C.
3. Use the formula for the electric field E due to a point charge at a distance r: E = (1/(4πϵ₀)) × (Q/r²), where ϵ₀ is the permittivity of free space, approximately 8.85×10−12 C²/(N·m²).
4. Substitute the values into the formula: Q = 92 × 1.6×10−19 C and r = 1.0×10−10 m. Calculate the electric field E using these values.
5. Ensure the units are consistent and check the calculation for any errors. The result will give you the magnitude of the electric field at the distance of the electrons.

Key Concepts

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the electric force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This principle is essential for calculating the electric field produced by a charged nucleus at a given distance.

Electric Field

The electric field is a vector field around a charged object where a force would be exerted on other charges. It is defined as the force per unit charge and is measured in newtons per coulomb (N/C). Understanding the electric field is crucial for determining the magnitude of the field produced by the uranium nucleus at the distance of the electrons.

Spherical Charge Distribution

A spherical charge distribution assumes that the charge is uniformly distributed over the surface of a sphere. For a spherically symmetric charge distribution, the electric field outside the sphere can be calculated as if all the charge were concentrated at the center. This concept simplifies the calculation of the electric field at a distance from the nucleus.