Question

A point charge Q is at a point (a,b,0) of a rectangular coordinate system. The halfplanes (x≥0,y=0) and (x=0,y≥0) are conducting. Determine the electric field at a point (x,y,0), where x>0 and y>0.

Answer 1

To determine the electric field at a point (x, y, 0), we need to consider the contributions from the point charge Q and the conducting halfplanes. We can use Coulomb's Law to calculate the electric field due to the point charge and use boundary conditions to determine the electric field due to the conducting halfplanes. By summing up these contributions, we can find the net electric field at the point (x, y, 0).

Step-by-step explanation:

To determine the electric field at a point (x, y, 0) in a rectangular coordinate system, we can consider the contributions from the point charge Q and the conducting halfplanes. The electric field due to a point charge can be calculated using Coulomb's Law, which states that the magnitude of the electric field at a point is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance between the point and the charge.

The electric field due to the conducting halfplanes can be determined using boundary conditions. At the boundary of a conductor, the electric field is perpendicular to the surface. In this case, the conducting halfplanes (x≥0, y=0) and (x=0, y≥0) would produce an electric field that is perpendicular to the x-axis and the y-axis, respectively. Therefore, the net electric field at the point (x, y, 0) can be calculated by summing the contributions from the point charge Q and the conducting halfplanes.

Let's break down the steps to calculate the electric field:

1. Calculate the electric field due to the point charge Q using Coulomb's Law.
2. Calculate the electric field due to the conducting halfplanes along the x-axis and the y-axis. For the x-axis, the electric field would be perpendicular to the x-axis and have a magnitude of zero at the point (x, y, 0). For the y-axis, the electric field would be perpendicular to the y-axis and have a magnitude of zero at the point (x, y, 0).
3. Add up the contributions from the point charge Q and the conducting halfplanes to get the net electric field at the point (x, y, 0).