Final answer:
To determine the magnitude of the electric field at an arbitrary distance r from the center of a uniformly charged insulating sphere, we can apply Gauss's law. The magnitude of the electric field is given by E = (4/3) * p * r / ε₀, where p is the uniform volume charge density, r is the distance from the center of the sphere, and ε₀ is the permittivity of free space. This expression holds true as long as r is less than the radius of the sphere.
Step-by-step explanation:
To determine the magnitude of the electric field at an arbitrary distance r from the center of a uniformly charged insulating sphere, we can apply Gauss's law. Gauss's law states that the total electric flux through a closed surface is proportional to the total charge enclosed by that surface. By applying Gauss's law to a spherical surface of radius r and finding the charge enclosed by that surface, we can determine the electric field. In this case, since we are considering a solid uniformly charged insulating sphere, the charge enclosed by the spherical surface will be the total charge of the sphere.
Since the sphere has a uniform volume charge density p, we can calculate the charge Q enclosed by a spherical surface of radius r by multiplying the volume of the sphere enclosed by the surface by the volume charge density:
Q = (4/3)πr³p
By applying Gauss's law to the spherical surface, the electric flux through the surface is given by Φ = E * A, where E is the magnitude of the electric field and A is the surface area of the spherical surface:
Φ = E * (4πr²)
Since the sphere has a uniform charge distribution, the electric field will also have a spherically symmetric distribution. Therefore, the electric field will only have a radial component (along the line connecting the center of the sphere and the point where the electric field is being observed):
E = E(r)*âr
Using Gauss's law, we can equate the electric flux through the spherical surface with the charge enclosed:
Φ = Q / ε₀
Substituting the expressions for Φ, Q, and A:
E * (4πr²) = (4/3)πr³p / ε₀
Simplifying the equation:
E = (4/3) * p * r / ε₀
Therefore, the magnitude of the electric field at an arbitrary distance r from the center of the uniformly charged insulating sphere, such that r < R, is given by E = (4/3) * p * r / ε₀.