Question

a negative 70 nC charge is distributed uniformly along the y-axis from -1.0m to 3.0m. what is the x-compenent of the electric field at x=2.0m on the x-axis? (answer is 101 N/C) I need help getting to this answer

Answer 1

Electric field due to a line charge is given by ,

lambda = 70/4 = 17.5 nC / m

z = 2

a = 1

b = 3

k = 9*10^9

substitue in Ez equation

Ez = (9*10^9*17.5*10^-9 / 2) * ( (3 / sqrt(4+9)) + (1/sqrt(4+1)))

Ez = 100.742 = 101 N/C

Answer 2

Let us take an element of length dy a distance y from the origin.

Charge on element=70dy/(3+1)=-17.5dy nC/m

x component of field due to this element at (2,0)= k*-17.5*10^-9 dy*(2/sqrt(2^2+y^2))/(2^2+y^2)= -35*10^-9kdy/(2^2+y^2)^1.5

The field is obtained by integrating from y=-1 to y=3.

E=$\int_{-1}^{3}-35*10^{-9}*9*10^9dy/(2^2+y^2)^{1.5}=-315\int_{-1}^{3}dy/(2^2+y^2)^{1.5}$=-100.75N/C~-101N/C