Find an expression for the electric field at a distance r from an infinitely long cylindrical rod having a uniform positive linear charge density
To find the expression for the electric field at a distance r from an infinitely long cylindrical rod, we can use Gauss's Law.
Gauss's Law states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space (ε₀).
We can consider a cylindrical Gaussian surface with radius r and length L, centered around the cylindrical rod. Since the rod is infinitely long, the length L can be chosen arbitrarily.
The electric field will be constant along the curved surface of the cylinder due to its symmetry. This allows us to simplify the calculation by considering only the flat ends.
Let's assume that the total charge on the rod is Q and the linear charge density is represented by λ, which is the charge per unit length.
We can find the charge enclosed by this Gaussian surface by multiplying the linear charge density (λ) by the length (L) of the Gaussian surface:
Qenc = λ * L
Now, let's calculate the electric flux through the ends of the cylinder. Since the electric field is constant and perpendicular to the surface, the flux through each end can be calculated as:
Flux = E * A
where A is the area of each end of the cylinder.
The area of each end of the cylinder is given by:
A = π * r^2
Using Gauss's Law, we have:
Flux = Qenc / ε₀
E * A = Qenc / ε₀
E * (π * r^2) = Qenc / ε₀
Since we know Qenc = λ * L, we can substitute this into the equation:
E * (π * r^2) = (λ * L) / ε₀
Now, solving for E, we get the expression for the electric field at a distance r from the infinitely long cylindrical rod:
E = λ / (2 * π * ε₀ * r)
Therefore, the expression for the electric field at a distance r from an infinitely long cylindrical rod with a uniform positive linear charge density (λ) is:
E = λ / (2 * π * ε₀ * r)