A 12-nC charge is distributed uniformly along the y axis from y = 0 to y = 4 m.Which of the following integrals is correct for the x component of the electric field at x = 2 m on the x axis?
E)none of these

Question

Quizplus · Accepted Answer

The electric field due to a small element of charge dy at a height y along the y-axis at a distance x on the x-axis is given by:
dE = k(dy/x^2)
The x component of this electric field is:
dEx = dE cosθ = dE(x/r) = dE(x/(x^2 + y^2)^(1/2))
Substituting the value of dE, we get:
dEx = k(dy/x^2) * (x/(x^2 + y^2)^(1/2))
The total electric field at point P(x=2m, y=4m) can be found by integrating this expression over the entire charge distribution along y-axis (0 to 4m):
E(x=2m) = ∫ dEx = k ∫(0 to 4m) (dy/x^2) * (x/(x^2 + y^2)^(1/2))
= k/x^2 ∫(0 to 4m) (y/(x^2 + y^2)^(1/2)) dy
= k/x^2 [-cosθ] (0 to 4m)
= k/x^2 [- (x/√(x^2 + y^2))] (0 to 4m)
= k/x^2 (-1/√17 + 1/√2)
= 0.1668 N/C
Using Coulomb's Law, we can also find the electric field at P(x=2m, y=4m) due to the entire charge distribution:
E = kQ/r^2 = k(12*10^-9)/(4^2 + 2^2) = 0.1668 N/C
Therefore, the correct integral for the x component of the electric field at x=2m is:
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A 12-NC Charge Is Distributed Uniformly Along the Y Axis