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Advanced Physics Homework Help
Applying the Biot-Savart Law to solenoids
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[QUOTE="RawrSpoon, post: 5008581, member: 440786"] Not really homework but I figured this was the best place to post anyway. [B] 1. Homework Statement [/B] I want to find the magnetic field B for an arbitrary solenoid using the Biot-Savart Law. I can find it easily through Ampere's Law, but I'd like mastery over the Biot-Savart Law. [h2]Homework Equations[/h2] [tex]B=\frac{μ_{0}}{4\pi}\int \frac{K \times (r-r')}{|r-r'|^3}da' [/tex] As there's no curly r as used in Griffiths Electrodynamnics, I'll replace curly r with an arbitrary symbol γ such that [tex]\gamma=r-r'[/tex] Thus [tex]\hat{\gamma}=\frac {r-r'}{|r-r'|}[/tex] And so [tex]B=\frac{μ_{0}}{4\pi}\int \frac{K \times \hat{\gamma}}{\gamma^2}da' [/tex] Maybe unnecessary but perhaps not. I just enjoy that notation as it's what I'm used to. [B] 3. The Attempt at a Solution [/B] As K is the surface charge density, I'll make the supposition that [tex]K=\frac{NI}{L}=nI[/tex] where N is the number of turns on the solenoid, and L is the length of said solenoid. And this is where I get stuck. Whereas I easily found the magnetic field due to a single loop of wire, the solenoid having length makes me be very unsure as to where to even begin. Whereas a loop of wire has γ easily defined as [tex]\gamma=\sqrt {R^2+x^2}[/tex] where R is the radius of the loop of wire and x is the distance of an arbitrary point M along the same axis as the loop I've found that γ at the part of the solenoid closest to M is equal to [tex]\gamma=\sqrt {R^2+x^2}[/tex] the part of the solenoid furthest from M gives a γ of [tex]\gamma=\sqrt {R^2+(x+L)^2}[/tex] I'm really lost. This doesn't mean I haven't tried my hardest or I'm being lazy. Any attempt to solve in a similar manner to that of a loop of wire ends up giving me multiple integrals that become a massive headache really quickly. Any nudge in the right direction would be greatly appreciated. If anything is unclear, let me know so I can try to either clarify my language or even attempt to draw a (likely very poor quality) image. [/QUOTE]
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Applying the Biot-Savart Law to solenoids
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