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This question is beyond me, and I have very little direction for solving it. I understand that the component of $\ B $ perpendicular to the $x$-axis is 0, and that the component parallel along the $x$-axis is doubled, which is where I assume the 2 on the denominator comes from.

I can't find anything in my lecture notes/online describing a general formula for the Biot-Savart Law for any point on the x-axis. Anyone have any ideas?

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Here is diagram 4)b) mentioned in the question:

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  • $\begingroup$ Is the definition not in the text used in your class (if you have one)? $\endgroup$
    – Triatticus
    Commented May 7 at 17:26

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The $\bf B$ field on the axis a distance $\bf d$ above a loop is given by \begin{eqnarray} {\bf B(d)} & = & \frac{I}{c}\oint \frac{[{\bf dr'}\times{\bf (r-r')}]}{|{\bf r-r'}|^3}\nonumber\\ & = &\frac{I}{c}\int_{0}^{2\pi}\frac{a^2d\theta{\bf\hat d}}{(a^2+d^2)^{\frac{3}{2}}} \nonumber\\ & = &\frac{2\pi I a^2{\bf\hat d}}{c(a^2+d^2)^{\frac{3}{2}}}. \label{bloop} \end{eqnarray} (In Gaussian units.)

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