You wish to determine the electric field magnitude along the perpendicular bisector of a 230-mm line along which35 nC of charge is distributed uniformly. You want to get by with a minimal amount of work, so you need to know when it is sufficient to approximate the line of charge as a charged particle.
At what distance along the perpendicular bisector does your error in E reach 5 % when you use this approximation?
The Attempt at a Solution
If I do the approximation part first, I get:
Plugging in the values (minus units at this time):
So now I work on the Eline-charge.
Because point P at distance d is along the bisector of the line, I know that Ex cancels itself out. All I need to worry about is Ey.
I punch that through an integral calculator, and I get
6. E=kλ[x/y(x2+y2)1/2] ---- evaluated from -L/2 to L/2
Now I start to plug in all the values (minus the units for the time being) and I get
Now I'm working on finding the approximate error.
I know the error (5%) and now I want to solve for y at that error.
This is pretty nasty, so I plugged it in the Mathway calculator and it promptly informed me there are no solutions.
That sums up what I've done. Where am I going wrong?