To determine the electric field strength on the axis of a charged circular disk, we need to analyze how the charge distribution affects the electric field at a point along the axis. In this case, the charge density varies with the radial distance from the center of the disk, which adds an interesting layer to our calculations.
Understanding the Charge Distribution
The surface charge density, denoted as σ, is proportional to the radial distance r from the center of the disk. This can be expressed mathematically as:
where k is a constant of proportionality. The total charge on an infinitesimal ring of radius r and thickness dr can be calculated as:
- dQ = σ(r) * dA = σ(r) * 2πr dr = k * r * 2πr dr = 2πk r² dr
Calculating the Electric Field
To find the electric field at a distance h along the axis of the disk, we consider the contribution of each infinitesimal ring of charge. The electric field due to a ring of charge at a distance h along the axis can be derived from Coulomb's law. The electric field dE produced by the ring at point P (located at distance h from the center) is given by:
- dE = (1 / (4πε₀)) * (dQ / (r² + h²)) * cos(θ)
Here, cos(θ) can be expressed in terms of h and r:
Substituting dQ into the equation for dE, we get:
- dE = (1 / (4πε₀)) * (2πk r² dr / (r² + h²)) * (h / √(r² + h²))
Integrating Over the Disk
To find the total electric field E at point P, we need to integrate dE from r = 0 to r = R (the radius of the disk):
- E = ∫(dE) = ∫(0 to R) (1 / (4πε₀)) * (2πk r² h / (r² + h²)√(r² + h²)) dr
This integral can be simplified and evaluated using standard calculus techniques. The result will yield an expression for the electric field strength E at the axial distance h from the center of the disk.
Final Expression for Electric Field Strength
After performing the integration, we arrive at the expression for the electric field strength on the axis of the disk:
- E = (k / 2ε₀) * (R² / (R² + h²)^(3/2)) * (h - √(R² + h²))
This formula shows how the electric field strength depends on the charge distribution, the radius of the disk, and the distance from the center along the axis. The presence of the square root and the cubic term in the denominator indicates how the electric field diminishes with increasing distance from the disk.
In summary, the electric field strength at a distance h from the center of a charged circular disk with a surface charge density proportional to the radial distance can be derived using principles of electrostatics and calculus. This approach not only highlights the relationship between charge distribution and electric field but also emphasizes the importance of integration in solving problems in electrostatics.
