Final answer:
The electric field at a point on the z-axis due to a charged ring in the x-y plane is found by integrating the contributions of each segment of charge, considering the ring's symmetry, so only the vertical component contributes to the net field.
Step-by-step explanation:
The student asks about the electric field created by a ring of line charge with a non-uniform charge density that varies with the angle φ. We're specifically interested in the electric field at a point located along the z-axis, at a height z above the x-y plane. To solve this, we use symmetry and integrate over the ring of charge to find the total field at the desired point. Given the charge density λ = 5 cos(φ/2) C/m and a point at z=3 cm above the center, the electric field along the z-axis can be found using Coulomb's law and calculus to integrate the contributions of each infinitesimal segment of charge on the ring.
Since the ring is symmetrical, the horizontal components of the electric field cancel out, and only the vertical component along the z-axis contributes to the net electric field at the point. By integrating the contributions, taking into account the linear charge density given and the geometry of the problem, one could find the magnitude and direction of the electric field at the point.