A long, thin rod parallel to the y-axis is located at x=-1.0 cm and carries a uniform linear charge density of + 1.0 nc/m. a second long, thin rod parallel to the z-axis is located at x=+1.0 cm and carries a uniform linear charge density of - 1.0 nc/m. what is the net electric field due to these rods at the origin? (ϵ0=8.85*10-12 c2/n · m2)

These long thin rods are essentially just infinite lines of charge, through gauss' law, we can get the electric field for an infinite line of charge to be E = 2Kλ/r, where E is the electric field, K is the electrostatic constant (same as 1 / (4πε₀) = 9*10⁹), λ is the linear charge density and r is the distance we are from the line of charge. We'll have 2 of these terms, one for each line of charge. Both lines of charge are 1 cm from the origin, so the r's drop out. Then one has a + 1 linear charge density and one has a - 1 linear charge density. The 2 and the K are constants and won't matter, so we have the electric field cancel out and we get an E field of 0 at the origin. Math looks like this E = 2Kλ/r + 2Kλ/r = 2K (+1) / (1) + 2K (-1) / (1) = 2K - 2K = 0