Find the electric field on the axis of a uniformly charged solid cylinder, at a distance z from the center of the cylinder. The length of the cylinder is H, its radius is “a”, and the charge density is ρ. (Assume that z > H/2).
To find the electric field on the axis of a uniformly charged solid cylinder at a distance z from the center, you can follow these steps:
Step 1: Identify the given parameters:
- H: length of the cylinder
- a: radius of the cylinder
- ρ: charge density
- z: distance from the center of the cylinder (assume z > H/2)
Step 2: Solve for the electric field using these steps:
1. Calculate the linear charge density (λ):
- λ = ρ * H
2. Break down the cylinder into infinitesimally small rings of thickness dx.
- The charge enclosed by each ring is given by dq = λ * dx
3. Express the position vector from each ring to the point on the axis where you want to calculate the electric field as r (distance vector).
- For a ring at a distance x from the axis, r = ((a^2 + z^2)^(1/2) - x)
4. Calculate the electric field contribution (dE) from each ring:
- dE = (k * dq) / r^2
- where k is the Coulomb's constant (k = 9 * 10^9 Nm^2/C^2)
5. Integrate the electric field contribution (dE) from all the rings to find the total electric field (E):
- Integrate dE from 0 to a:
- E = ∫ dE = ∫ (k * dq) / r^2
- Substitute dq = λ * dx and r = ((a^2 + z^2)^(1/2) - x)
6. Evaluate the integral and simplify to obtain the final expression for the electric field on the axis of the cylinder at distance z:
- E = (k * λ / a) * ln((a + ((a^2 + z^2)^(1/2))) / z)
The final expression provides the electric field on the axis of the uniformly charged solid cylinder at a distance z from the center.