A continuous line of charge lies along the x axis, extending from x = +x0 to positive infinity. The line carries positive charge with a uniform linear charge density λ0. (a) What is the magnitude of the electric field at the origin? (Use the following as necessary: λ0, x0, and ke.) E = (b) What is the direction of the electric field at the origin? +x −x +y −y +z −z

 E = k λ₀ / x₀, the field is in thenegative direction of the x axis (-x)

Explanation:

In this problem the electric field of a line of charge is requested, the expression for the electric field is

          E = k ∫ dq / r²

where k is the Coulomb constant that you are worth 9 10⁹ N m²/C², that the charge and r the distance to the point of interest, in this case it is the origin (x = 0)

let's use the definite linear density

        λ₀ = dq / dx

        dq = λ₀ dx

we replace and integrate

       E = k λ₀ ∫ dx / x²

       E = k λ₀ ( -1 / x)

we evaluate the integral from the lower limit of load x = x₀ to the upper limit x = ∞

       E = - k λ₀ (1 /∞ - 1 / x₀)

       E = k λ₀ / x₀

as the field is positive the direction is away from the charges, so it is in the negative direction of the x axis (-x)

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west