Deriving expression for magnetic field at point P due to current

[jisbon]

Homework Statement

Derive an expression for the magnetic field at point P due to current-carrying wire of length a

Relevant Equations

NIL

So I think I do understand how to do this, but the thing is my answers are always incorrect. Will need some guidance/help on this.
##B =\frac{\mu_{0}I}{4\pi}\int\frac{dysin\theta}{r^2} ##
##y = Rtan\phi##
##dy = Rsec^2\phi d\phi##
##B = \frac{\mu_{0}I}{4\pi}\int\frac{dysin\theta}{r^2} = \frac{\mu_{0}I}{4\pi}\int\frac{Rsec^2\phi d\phi sin(\frac{\pi}{2}-\phi)}{R^2sec^2 \phi}
= \frac{\mu_{0}I}{4\pi}\int\frac{cos\phi}{R}d\phi = \frac{\mu_{0}I}{4\pi}(sin\phi_{2}-sin\phi_{1}) ##

So,
##sin\phi_{1}=\frac{2/3a}{\sqrt{4/9a^2+x^2}}##
##sin\phi_{2}=\frac{1/3a}{\sqrt{1/9a^2+x^2}} ##
##\frac{\mu_{0}I}{4\pi}(sin\phi_{2}-sin\phi_{1}) = \frac{\mu_{0}I}{4\pi} (\frac{1/3a}{\sqrt{1/9a^2+x^2}} - \frac{2/3a}{\sqrt{4/9a^2+x^2}})##

Is this correct?

 

[Delta2]

I think it is correct, just a typo you forgot the term ##\frac{1}{x}## or ##\frac{1}{R}## so the correct formula should be
$$B(x)=\frac{\mu_0 I}{4\pi x}(\frac{a}{3\sqrt{x^2+\frac{a^2}{9}}}-\frac{2a}{3\sqrt{x^2+\frac{4a^2}{9}}})$$One rule of thumb to check such expressions (for static electric or magnetic fields) is that for distances far away from the source (that must have finite dimensions) , you should get approximately an inverse square law, and such is the case here, for ##x## large (in comparison with ##a##) the ##x^2## term dominates in the square roots so each square root approximately simplifies to ##\sqrt{(x^2+0)}=x## and together with the other ##x## from outside the parenthesis we get an inverse square law.