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## Homework Statement

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?

## Homework Equations

dE=kλdx/r

^{2}

E=kQ/r

^{2}

λ=Q/L

## The Attempt at a Solution

If I do the approximation part first, I get:

1. E

_{approx}=kQ/y

^{2}

Plugging in the values (minus units at this time):

2. E

_{approx}=315/y

^{2}

So now I work on the E

_{line-charge}.

Because point P at distance d is along the bisector of the line, I know that E

_{x}cancels itself out. All I need to worry about is E

_{y}.

3. dE=((kλdx)/(x

^{2}+y

^{2})

^{2})*(y/(x

^{2}+y

^{2})

^{1/2})

4. dE=(kλydx)/(x

^{2}+y

^{2})

^{3/2})

=>

5. E=kλ∫(ydx/(x

^{2}+y

^{2})

^{3/2}

I punch that through an integral calculator, and I get

6. E=kλ[x/y(x

^{2}+y

^{2})

^{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

7. E=1370[0.23/(y(.013+y

^{2})

^{1/2})]

8. E=315.1/(y(.013+y

^{2})

^{1/2})

**Okay.**

Now I'm working on finding the approximate error.

error=(|E

_{approx}-E

_{exact}|)

**/**E

_{exact}

I know the error (5%) and now I want to solve for y at that error.

9. 0.05=|315/y

^{2}-315.1/(y(.013+y

^{2})

^{1/2})|

**/**315.1/(y(.013+y

^{2})

^{1/2})

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?

Thanks,

SOG