Question

Charge of uniform linear density 3.0 nC/m is distributed along the x axis from x = 0 to x = 3 m. Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +4 m on the x axis?

1) ∫[0,4] (3.0 nC/m) / (4πε₀(x - 4)) dx
2) ∫[0,4] (3.0 nC/m) / (4πε₀(4 - x)) dx
3) ∫[0,4] (3.0 nC/m) / (4πε₀(4 + x)) dx
4) ∫[0,4] (3.0 nC/m) / (4πε₀(x + 4)) dx

Answer 1

None of the given options are correct for calculating the electric potential at x = +4 m due to a charge distributed from x = 0 to x = 3 m. The closest option to the correct answer is option 2, with an upper limit of integration adjusted to 3.

Step-by-step explanation:

The correct integral to calculate the electric potential at the point x = +4 m on the x-axis due to a charge of uniform linear density 3.0 nC/m distributed along the x-axis from x = 0 to x = 3 m is given by:

∠[0,3] (3.0 nC/m) / (4πε₀(4 - x)) dx.

If the charge is distributed along the x-axis from x = 0 to x = 3 m, then the limits of integration should be from 0 to 3, not 0 to 4 as stated in the question options. The distance from a point x on the axis to the point at x = 4 m is (4 - x), so the denominator of the integrand should be (4 - x). Therefore, none of the given options are correct. However, the closest option that represents the correct form of the electric potential integral is option 2, except that the upper limit of the integral should be 3 instead of 4.