Show Fx,Fy when x>>a: 65 Character Title"Show Fx,Fy When x Much Greater Than a

[baseballman]

Show that if x>>a...

Homework Statement

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)

Homework Equations

I need to show that when x>>a:

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

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

 

[WiFO215]

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

 

[CompuChip]

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.

 

[baseballman]

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.

 

[ehild]

Do you know limits?

ehild

 

[WiFO215]

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.

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!}...

You can read more about Taylor series here:
http://en.wikipedia.org/wiki/Taylor_series

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 F_y properly.