State only Gauss law in integral form.Use it to find out electric field due to an isolated homogeneous sphere of charges at an outside point.
http://farside.ph.utexas.edu/teaching/302l/lectures/node30.html
Gauss's law in integral form states that the electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space, ε₀.
Mathematically, Gauss's law in integral form can be written as:
∮ E · dA = (1/ε₀) ∫ ρ dV
Where:
∮ E · dA represents the surface integral of the electric field E dotted with the differential area vector dA over a closed surface,
ρ represents the charge density (charge per unit volume), and
∫ ρ dV represents the volume integral of the charge density ρ over the volume enclosed by the closed surface.
Now, let's use Gauss's law to find the electric field due to an isolated homogeneous sphere of charges at an outside point.
Consider a sphere of radius R centered at the origin and let's assume it carries a total charge Q uniformly distributed.
To apply Gauss's law, we choose a closed Gaussian surface that is a sphere of radius r > R, centered at the origin. The surface integral of the electric field over this Gaussian surface will give us the electric flux through it.
Since the electric field is radially symmetric for a uniform sphere, the electric field E will be directed radially outward at every point on the Gaussian surface.
By symmetry, the electric field magnitude will be constant on the Gaussian surface. So, we can take it out of the surface integral and write:
|E| ∮ dA = (1/ε₀) ∫ ρ dV
The left-hand side of the equation represents the surface area of the sphere, which is 4πr², multiplied by the magnitude of the electric field.
The right-hand side of the equation represents the total charge enclosed by the Gaussian surface, which is Q, divided by ε₀.
Therefore, we can write the equation as:
|E| * 4πr² = (1/ε₀) * Q
Rearranging the equation to solve for the magnitude of the electric field, we get:
|E| = (Q / (4πε₀r²))
So, the electric field magnitude at an outside point due to an isolated homogeneous sphere of charges is given by:
E = (Q / (4πε₀r²)) * r̂
Here, r̂ is a unit vector in the radial direction from the origin to the outside point.