Evaluating a simple electrostatic field integral

[Taulant Sholla]

Homework Statement

I can find the e-field at point P.

Homework Equations

I get, easily enough, the correct integral solution (for the y-component, E_y - which is all I need to do):

which I can see, informally, evaluates to:

which is the correct answer.

The Attempt at a Solution

My question: is it, technically, sufficient for me to simply explain: "well, when evaluated, the infinity in the numerator cancels with sqrt(infinity^2) in the denominator? Or can I offer a more rigorous explanation?

I tried L'Hôpital's Rule, but it doesn't seem to lead anywhere as far as showing anything definitive.

Thank you - any hints are appreciated!

 

[blue_leaf77]

Try factoring out $x^2$ from the squareroot in the denominator, you should know how the resulting $R^2/x^2$ behaves when $x \to \infty$.

 

[Taulant Sholla]

Hmmm (thanks!). Yes, R²/x² disappears as x→∞, but that now leaves an un-cancelled x in the numerator that is going to infinity.

 

[rude man]

Hmmm (thanks!). Yes, R²/x² disappears as x→∞, but that now leaves an un-cancelled x in the numerator that is going to infinity.

You're not factoring the denominator correctly.
(x² + R²)^1/2 = {x²(1 + R²/x²)}^1/2
= x(1 + R²/x²)^1/2
etc.

 

[Taulant Sholla]

Ach, yes. Sorry - and thank you!

 

[rude man]

Ach, yes. Sorry - and thank you!

Kein Problem!