Final answer:
To find the electric potential on the axis of a disk with charge Q, the charge density σ is determined and then an integral is computed, which results in a formula involving the radius R, the axial distance z, and the permittivity of free space ε_0.
Step-by-step explanation:
To calculate the electric potential on the axis of a disk due to a uniformly distributed charge Q, consider each infinitesimally small charge dq as a point charge and integrate the potential contribution from each dq along the disk's surface. The uniform charge distribution is represented as σ = Q / (π R^2), where σ is the charge density, Q is the total charge, and R is the radius of the disk.
The electric potential V at a point at a distance z along the axis can be expressed as V = (1 / (4πε_0)) ∫ σ dq / r, where r is the distance from the point to the charge element dq, ε_0 is the vacuum permittivity, and the integration is performed over the entire disk surface. Because r varies with the radial distance from the axis, it can be related to z and the radial distance ρ of dq by the Pythagorean theorem: r^2 = z^2 + ρ^2.
The integral is then set up in cylindrical coordinates with ρ ranging from 0 to R, and the result to compute the potential at the point on the axis is given by the formula V = σ / (2ε_0) · (z - √(z^2 + R^2)). When σ is replaced with Q / (πR^2), the final expression for V becomes V = Q / (2πε_0R^2) · (z - √(z^2 + R^2)).