Gauss's Law and Parallel Line Charges

[goober]

Homework Statement

Find an equation for the net electric field at a point, above and between, two infinite line charges, one with line charge density λ and the second with line charge density -λ. The point is a distance R from both line charges, a distance y above the midpoint between charges, and the distance between charges is 2x. For reference, if the lines are viewed as parallel to the z-axis in an xy-plane, the line charges and the point would form a triangle with base 2x and height y.

Homework Equations

$\vec E$ = (λ / 2π∈₀r)$\hat r$ (line charge field from Gauss's Law)
$\hat r$ = $\vec r$ / r
R = (x² + y²)^1/2

The Attempt at a Solution

$\vec E$₁ = (λ / 2π∈₀R) [ (x$\hat i$ + y$\hat j$) / R ]
$\vec E$₂ = (-λ / 2π∈₀R) [ (-x$\hat i$ + y$\hat j$) / R ]
$\vec E$_net = $\vec E$₁ + $\vec E$₂ = 2xλ / 2π∈₀R²$\hat i$ = (xλ / π∈₀R²)$\hat i$

I found the electric field for each line charge individually using the equation given in the textbook for Gauss's law on an infinite line charge. I then added the two field vectors and found a final result where the y-components canceled and the x-components doubled. I just need to know if this is the correct method for solving the problem, because I was led to believe it would be more complex.

 

[kuruman]

Hi goober and welcome to PF.

I just need to know if this is the correct method for solving the problem, because I was led to believe it would be more complex.

That's it. It's known as superposition.