Question

Consider a real-world simplified Hydrogen atom case, which has a positive static charge e₀ at (0,0,0) location and a negative charge -e₀ with charge density spatial distribution of rho(r) = -e₀/πa³.e⁻²ʳ/ᵃ. a is radius of the electron circulation path. Now using Gaussian law ∇.E = rho/e₀ to calculate the spatial E field distribution in spherical coordinates.

Answer 1

To find the electric field distribution around a simplified Hydrogen atom, one must apply Gauss's law considering the spherical symmetry of the charge distribution. By using a spherical Gaussian surface and integrating the given charge density, the electric field both inside and outside the charge distribution can be obtained.

Step-by-step explanation:

The question asks to calculate the electric field distribution around a simplified Hydrogen atom using Gaussian law. The charge density, ρ(r), is given as -e₀/πa³.e⁻²⁰/⁰. This density is a function of the radial distance 'r' from the center of the atom, and 'a' represents the radius of the electron circulation path. Gauss's law states that the divergence of the electric field, ∇.E, is equal to the charge density ρ divided by the vacuum permittivity e₀. When applying this law in spherical coordinates, the electric field can be calculated by integrating over a closed spherical Gaussian surface concentric with the charge distribution.

Strategy for solving:

1. Assume spherical symmetry and represent the field as É = E(r)î.
2. Apply Gauss's law over a closed spherical Gaussian surface S of radius r, concentric with the charge distribution.
3. Integrate the charge density within the Gaussian surface to find the total enclosed charge.
4. Use the symmetry to simplify the expression for electric flux through the Gaussian surface.
5. Solve for the electric field E(r) at points inside and outside the spherical distribution.

The electric field inside a spherically symmetric charge distribution depends on the charge enclosed by the Gaussian surface up to radius 'r', while outside the distribution, it depends on the total charge of the distribution.