A circular disc of radius 5m with a surface charge density ρs = 10sinφ is enclosed by surface. What is the net flux crossing the surface?

The net electric flux, (Phi), crossing a surface that encloses a charge is given by Gauss’s law. Gauss’s law states that the flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space ((epsilon_0)). Mathematically, Gauss’s law is expressed as:

[

Phi = frac{Q_{text{enc}}}{epsilon_0}

]

The charge density is given as a function of (phi), (rho_s = 10sinphi), where (rho_s) represents the surface charge density. The total charge enclosed, (Q_{text{enc}}), can be calculated by integrating the charge density over the entire disc. The surface area element for a disc in polar coordinates is (dA = r,dr,dphi), where (r) is the radius and (phi) is the polar angle. However, since the charge density is given as a function of (phi) without dependence on (r), and considering the circular symmetry and the nature of the sine function, we can analyze this considering the charge distribution’s nature over ([0, 2pi]) range of (phi).

For a disc with a radius of 5m, integrating the charge density over the surface should account for the entire area, but the charge density varies with the angular component