What is the expression for the electric field inside a charged cylinder at a distance r from the axis? And what about the electric field at a point outside the cylinder in terms of the charge per unit length?
Electric Field Inside the Cylinder: E = (ρ * r) / (2 * ε₀)
Electric Field Outside the Cylinder: E = λ / (2 * π * R * ε₀)
Derivation of Electric Field Inside the Cylinder
The charge density within the cylinder can be calculated using the formula ρ = Qt / Vt, where Qt is the total charge on the cylinder and Vt is the net volume of the cylinder. By considering a cylindrical Gaussian surface enclosing the charge, the electric field inside the cylinder at a distance r from the axis can be derived.
From the given information, the charge enclosed within the Gaussian surface can be expressed as Q = ρV, where Q is the total charge within the surface and V is the volume enclosed. By integrating the electric field over the surface area of the cylinder, we find:
∮ Eds = q / ε₀
Substituting the values for surface area and charge with E (2πrl) and ρπr²l, respectively:
E(2πrl) = (ρπr²l) / ε₀
Rearranging the equation, we get:
E = (ρ * r) / (2 * ε₀)
Calculation of Electric Field Outside the Cylinder
Using the same integral form for the electric field, we can find the expression for the electric field outside the charged cylinder. By considering the total charge per unit length of the cylinder (λ), the electric field outside the cylinder can be determined.
Again, by integrating the electric field over a cylindrical Gaussian surface enclosing the charge, we have:
∮ Eds = Qt / ε₀
Substituting E (2πrl) for the integral and Qt for q, we find:
E(2πrl) = Qt / (2πRlε₀)
Replacing Qt with λl, where λ is the charge per unit length:
E = λ / (2 * π * R * ε₀)
Final Explanation:
To summarize, the electric field inside a charged cylinder at a distance r from the axis can be calculated using the charge density (ρ) and the distance (r) from the axis of the cylinder. On the other hand, the electric field at a point outside the cylinder is determined by the total charge per unit length (λ) along the cylinder's axis and the distance from the center of the cylinder. These derived expressions help in understanding the distribution of electric fields in cylindrical charge configurations.