Magnetic field at a point due to a line of charge

[vysero]

Homework Statement

A wire carrying a current I in the positive x direction is located along the x axis. The wire is of finite length and is located between x = -L and x = L. Find the magnetic field at a field point located a distance a away from the wire.

Homework Equations

Biot-Savart law
Ampere's law

The Attempt at a Solution

[/B]
I solved the problem above and its the same as the professors so its correct. However, I was following along with a video I found on lassevrien's channel on YouTube:

and he attained a different answer for what I believe is the same problem. If you don't want to watch the video I have summarized it on the picture (lower right); he is using Ampere's law.

So I figure if I evaluated my answer from -∞ too ∞ I would get what he got but I am getting an undefined answer. Are these two methods of doing the same problem? Am I just evaluating wrong?

 

[BvU]

and its the same as the professors so its correct

That is in general a very dangerous assumption: to err is human and professors are human

I am getting an undefined answer

How so ?

 

[vysero]

How so ?

I believe B would be undefined here:

$$B =\frac{μI}{4πa} (\frac {∞} {\sqrt{a^2 +∞^2}}\frac {∞} {\sqrt{a^2 +∞^2}}),$$

 

[BvU]

Divide numerator and denominator by $L$ before letting $L$ go off to $\infty$

 

[vysero]

Divide numerator and denominator by $L$ before letting $L$ go off to $\infty$

I am not sure if this is what you meant but here goes:

$$\lim_{x \rightarrow +\infty} {\frac {x} {\sqrt{a^2 +x^2}}}$$
$$\lim_{x \rightarrow +\infty} {\frac {(\frac{1}{x})x} {\sqrt{\frac{1}{x^2}}\sqrt{a^2 +x^2}}}$$
$$\lim_{x \rightarrow +\infty} {\frac {1} {\sqrt{(\frac{1}{x^2})(a^2 +x^2)}}}$$
$$\lim_{x \rightarrow +\infty} {\frac {1} {\sqrt{\frac{a^2}{x^2}}+1}}=1$$

 

[BvU]

Brilliant ! Now add 1 and 1 .

Well, almost brilliant: not $$ \lim_{x \rightarrow +\infty} {\frac {1} {\sqrt{\frac{a^2}{x^2}}+1}}=1$$ but $$
\lim_{x \rightarrow +\infty} {\frac {1} {\sqrt{{\frac{a^2}{x^2}}+1}}}=1$$

 

[vysero]

Got it, thanks man.

 

[BvU]

You are welcome.