A thin rod of finite length L has a uniform charge density X. Follow the steps below to show that the electric field E at a point P from the perpendicular bisector and lying at a distance Y is given by AL E = 4TE0YVL2 + 4y² Note that the (x, y) axes are arranged such that the origin (where a = 0 and y = 0) is coincident with the point at the middle of the rod and that x and y increase toward the right and toward the top, respectively. Also note how 0 and r are defined in the figure as well. a) What is dg in terms of A and dx? Express cos e and sin 0 in terms of y, r and r. Expres r in terms of x and y. Describe the symmetry of the charge distribution and therefore why the x-component of E is zero while the y-component ofE is non-zero. b) Express dEy in terms of solely x, y, and dx (along with 4, eo, A and T). c) Integrate your expression for dEy over x to obtain the expression for Ey = E given above. A useful integral identity here is (where a is a constant) %3D %3D dx (r2 + a2)3/2 a272 + a?

A thin rod of finite length L has a uniform charge density X. Follow the steps below to show that the electric field E at a point P from the perpendicular bisector and lying at a distance Y is given by AL E = 4TE0YVL2 + 4y² Note that the (x, y) axes are arranged such that the origin (where a = 0 and y = 0) is coincident with the point at the middle of the rod and that x and y increase toward the right and toward the top, respectively. Also note how 0 and r are defined in the figure as well. a) What is dg in terms of A and dx? Express cos e and sin 0 in terms of y, r and r. Expres r in terms of x and y. Describe the symmetry of the charge distribution and therefore why the x-component of E is zero while the y-component ofE is non-zero. b) Express dEy in terms of solely x, y, and dx (along with 4, eo, A and T). c) Integrate your expression for dEy over x to obtain the expression for Ey = E given above. A useful integral identity here is (where a is a constant) %3D %3D dx (r2 + a2)3/2 a272 + a?