Consider an infinitely long solid cylinder with radius R_0 and volume charge density rho=rho_0*r(r≤R_0) where rho_0 is a constant. Use first principles to determine the electric field E(vector) for r<R_0
To determine the electric field E(vector) for r < R_0 using first principles, we can use the principle of superposition.
First, let's consider a small element of the cylinder with volume charge density dρ. The charge contained within this element can be defined as dQ = dV * dρ, where dV is the differential volume of the element.
The electric field dE due to this small element at a point P, a distance r from the axis of the cylinder, can be given by Coulomb's Law:
dE = (k * dQ) / r^2
where k is the Coulomb's constant.
Now, let's derive the expression for dV in terms of r and dr.
The volume of a cylindrical shell with radius r and thickness dr can be given as dV = A * dr, where A is the area of the shell.
The area A can be calculated as A = 2πrh, where h is the height of the shell.
For a cylindrical element, h is approximately equal to the circumference of the cylinder, which is 2πr.
Therefore, A = 2πr * 2πr = 4π^2 r^2.
Substituting this back into the expression for dV, we get:
dV = 4π^2 r^2 dr
Now, let's substitute dV and dQ into the expression for dE:
dE = (k * dQ) / r^2
Substituting dQ = dV * dρ, we get:
dE = (k * dV * dρ) / r^2
Plugging in the expression for dV, we have:
dE = (k * 4π^2 r^2 dr * dρ) / r^2
Simplifying the equation, we get:
dE = 4π^2 k r dρ dr
Now, we need to integrate this expression over the entire volume of the cylinder. Since the cylinder is infinitely long, we consider the electric field only within the cylinder (r < R_0).
The total electric field E at a point P within the cylinder can be given by integrating the expression for dE over the volume of the cylinder:
E = ∫[0 to R_0] 4π^2 k r dρ dr
Here, we have the density function ρ = ρ_0 * r.
Substituting this into the expression, we get:
E = ∫[0 to R_0] 4π^2 k r (ρ_0 * r) dr
Evaluating this integral gives us the electric field E(vector) for r < R_0.