Question: Electric Field from a Right Triangle with Non$-$Uniform Charge Distribution

Consider a right triangle with vertices at the points $(0,0),(1,0),$ and $(0,1)$ in the Cartesian plane. The sides of the triangle are charged with a linear charge density given by $(x,y)=xy\frac{C}{m}.$

Tasks:

Charge Distribution Analysis:

Describe how the charge density $(x,y)=xy$ varies along each side of the triangle. Discuss the implications of this non$-$uniform distribution on the electric field generated by each side.

Electric Field Calculation:

Calculate the electric field contribution from each side of the triangle at the centroid of the triangle, which is located at the point $(\frac{1}{3},\frac{1}{3}).$

Superposition Principle:

Using the principle of superposition, determine the total electric field at the centroid by summing the contributions from each side of the triangle.

Discussion:

Discuss the challenges and considerations involved in calculating the electric field from a nonuniform charge distribution. How does the symmetry $($or lack thereof$)$ of the charge distribution affect the resultant electric field?

Instructions:

Provide detailed calculations for each part, showing all steps clearly.

Use diagrams to illustrate the charge distribution and the electric field vectors at the centroid.

Discuss any assumptions made during the calculations and their potential impact on the results.

Explain the physical significance of the results obtained for the electric field at the centroid.