Use Gauss' Law to prove that the electric field a distance r away from the center of a spherical charge Q with radius a is equal to kQr/a³. Note that a > r for this problem. Hint: Think about how much of the charge Q is actually "enclosed" for this situation, and that charges are uniformly distributed everywhere inside the insulqating sphere.
Gauss's law of electrostatics states that the electric flux, Φ, through any closed surface is proportional to the total electric charge enclosed within the surface, Q:
Φ = ∫ E . dA = Q/ε₀
Where E is the electric field, dA is the surface element, Q is the total charge enclosed by the surface, and ε₀ is the permittivity of free space.
This equation implies that the electric field must be perpendicular to the surface and that the electric flux is proportional to the charge enclosed within the surface. The law is useful for calculating the electric field for symmetrical charge distributions since symmetry can be used to simplify the calculation of the electric flux.
Charge density is the amount of electric charge per unit volume or unit area of a charged object or system. It is denoted by the symbol ρ and is measured in units of coulombs per cubic meter (C/m³) or coulombs per square meter (C/m²).
For a spherical charge distribution of total charge Q and radius a, the charge density is given by
ρ = Q/((4/3)πa^3) = (3Q)/4πa^3
where π is the mathematical constant pi. This equation expresses the charge density as the ratio of the total charge to the volume of the sphere. It allows us to determine the distribution of electric charge within the sphere, which is useful in calculating the electric field due to the sphere using Gauss's law. The charge density of a spherical charge distribution can also be used to determine the potential energy stored in the electric field surrounding the sphere.
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