Electric field by infinite line of charge

[Sam99]

Homework Statement

Given an infinitely long line of charge density λ extended along the x-axis, what is the electric field at a point X = x(x')+y(y')+z(z') (in space)?

Homework Equations

E = kq / r^2, dq = (lamda)dx

The Attempt at a Solution

dE = kλ ∫ [x(x')+y(y')+z(z')- x(x')] dx / [(y^2) +(z^2)]^(3/2) (integral from -∞ to +∞)

I end up getting 0 because -∞ + ∞ equals 0?

 

[tiny-tim]

Welcome to PF!

Hi Sam99! Welcome to PF!

(try using the X² button just above the Reply box )

dE = kλ ∫ [x(x')+y(y')+z(z')- x(x')] dx / [(y²) +(z²)]^3/2 (integral from -∞ to +∞)

shouldn't there be an x² on the bottom also?

 

[Sam99]

Thanks!
I don't think so because I'm using (x-x')/ |x-x'|^3.
would that be right?

 

[tiny-tim]

yes, but there's still a difference in x, which has to go on the bottom

 

[asteriatic]

Your solution is close, but you are missing a key component. The electric field at a point due to an infinite line of charge is given by the equation E = (λ / 2πε0) * (1 / r), where r is the distance from the point to the line of charge. This equation takes into account the fact that the electric field is not constant along the line of charge, but instead decreases as you move further away from the line.

To solve for the electric field at a point (x, y, z), you would need to integrate this equation along the entire length of the line, taking into account the changing distance as you move along the line. This integral would give you the total electric field at that point due to the entire line of charge.

In your attempt at a solution, you correctly set up the integral, but you need to use the correct equation for the electric field at a point due to an infinite line of charge. You also need to integrate over the entire length of the line, not just from -∞ to +∞. Once you have the correct integral set up, you can solve for the electric field at any point along the line of charge.