# Show that if x a

1. Mar 21, 2010

### baseballman

Show that if x>>a....

1. The problem statement, all variables and given/known data

I've got the following force generated by an electric field

$${F_x} = - KQq\frac{1}{{x\sqrt {{x^2} + {a^2}} }}$$

$${F_y} = \frac{{KQq}}{a}\left( {\frac{1}{x} - \frac{1}{{\sqrt {{x^2} + {a^2}} }}} \right)$$

2. Relevant equations

I need to show that when x>>a:

$${F_x} = - \frac{{KQq}}{{{x^2}}}$$

$${F_y} = \frac{{KQqa}}{{2{x^3}}}$$

3. The attempt at a solution

I think I'm on the right track but I'm stuck here:

$$\frac{1}{{\sqrt {{x^2} + {a^2}} }} = \frac{1}{x}{\left( {1 + \frac{{{a^2}}}{{{x^2}}}} \right)^{ - \frac{1}{2}}}$$

2. Mar 21, 2010

### WiFO215

Re: Show that if x>>a....

Have you heard of Taylor series? Why don't you try expanding the above term using taylor series and see which terms are negligible?

3. Mar 21, 2010

### CompuChip

Re: Show that if x>>a....

You are indeed on the right track. When x >> a, then $\delta := \frac{a}{x}$ is very small. So you can expand
$$x \sqrt{1 + a^2 / x^2} = x \sqrt{1 + \delta^2}$$ around $\delta = 0$.

4. Mar 21, 2010

### baseballman

Re: Show that if x>>a....

Thanks for your responses anirudh215 and CompuChip, but I really don't know what you mean by "expand". If you could show some steps I might have a clue.

5. Mar 21, 2010

### ehild

Re: Show that if x>>a....

Do you know limits?

ehild

6. Mar 21, 2010

### WiFO215

Re: Show that if x>>a....

Certain functions can be "expanded" around points, i.e. if you have a function f, and it is "expandable", then you can equate the value of that function about a point in terms of its derivatives.

The square root function above is one such function which is expandable. The Taylor series would be something like

$$\sqrt{1 + x} = 1 + \frac{x}{2} - \frac{x^{2}}{4.2!}$$....

Now, substitute $$\frac{a^{2}}{x^{2}}$$ as y, and expand similar to above. See what you can do from here. Note: the number of terms I have used in the expansion above is enough for you to complete the sum. Just look at what is and isn't negligible. I'm just doubtful whether you calculated Fy properly.