- #1
RawrSpoon
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Not really homework but I figured this was the best place to post anyway.
1. Homework Statement
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.
B=\frac{μ_{0}}{4\pi}\int \frac{K \times (r-r')}{|r-r'|^3}da'
As there's no curly r as used in Griffiths Electrodynamnics, I'll replace curly r with an arbitrary symbol γ such that
\gamma=r-r'
Thus
\hat{\gamma}=\frac {r-r'}{|r-r'|}
And so
B=\frac{μ_{0}}{4\pi}\int \frac{K \times \hat{\gamma}}{\gamma^2}da'
Maybe unnecessary but perhaps not. I just enjoy that notation as it's what I'm used to.
3. The Attempt at a Solution
As K is the surface charge density, I'll make the supposition that
K=\frac{NI}{L}=nI
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
\gamma=\sqrt {R^2+x^2}
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
\gamma=\sqrt {R^2+x^2}
the part of the solenoid furthest from M gives a γ of
\gamma=\sqrt {R^2+(x+L)^2}
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.
1. Homework Statement
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.
Homework Equations
B=\frac{μ_{0}}{4\pi}\int \frac{K \times (r-r')}{|r-r'|^3}da'
As there's no curly r as used in Griffiths Electrodynamnics, I'll replace curly r with an arbitrary symbol γ such that
\gamma=r-r'
Thus
\hat{\gamma}=\frac {r-r'}{|r-r'|}
And so
B=\frac{μ_{0}}{4\pi}\int \frac{K \times \hat{\gamma}}{\gamma^2}da'
Maybe unnecessary but perhaps not. I just enjoy that notation as it's what I'm used to.
3. The Attempt at a Solution
As K is the surface charge density, I'll make the supposition that
K=\frac{NI}{L}=nI
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
\gamma=\sqrt {R^2+x^2}
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
\gamma=\sqrt {R^2+x^2}
the part of the solenoid furthest from M gives a γ of
\gamma=\sqrt {R^2+(x+L)^2}
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.
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