Graduationwoot

Dragon Notes

i

\( \newcommand{bvec}[1]{\overrightarrow{\boldsymbol{#1}}} \newcommand{bnvec}[1]{\overrightarrow{\boldsymbol{\mathrm{#1}}}} \newcommand{uvec}[1]{\widehat{\boldsymbol{#1}}} \newcommand{vec}[1]{\overrightarrow{#1}} \newcommand{\parallelsum}{\mathbin{\|}} \) \( \newcommand{s}[1]{\small{#1}} \newcommand{t}[1]{\text{#1}} \newcommand{tb}[1]{\textbf{#1}} \newcommand{ns}[1]{\normalsize{#1}} \newcommand{ss}[1]{\scriptsize{#1}} \newcommand{vpl}[]{\vphantom{\large{\int^{\int}}}} \newcommand{vplup}[]{\vphantom{A^{A^{A^A}}}} \newcommand{vplLup}[]{\vphantom{A^{A^{A^{A{^A{^A}}}}}}} \newcommand{vpLup}[]{\vphantom{A^{A^{A^{A^{A^{A^{A^A}}}}}}}} \newcommand{up}[]{\vplup} \newcommand{Up}[]{\vplLup} \newcommand{Uup}[]{\vpLup} \newcommand{vpL}[]{\vphantom{\Large{\int^{\int}}}} \newcommand{lrg}[1]{\class{lrg}{#1}} \newcommand{sml}[1]{\class{sml}{#1}} \newcommand{qq}[2]{{#1}_{\t{#2}}} \newcommand{ts}[2]{\t{#1}_{\t{#2}}} \) \( \newcommand{ds}[]{\displaystyle} \newcommand{dsup}[]{\displaystyle\vplup} \newcommand{u}[1]{\underline{#1}} \newcommand{tu}[1]{\underline{\text{#1}}} \newcommand{tbu}[1]{\underline{\bf{\text{#1}}}} \newcommand{bxred}[1]{\class{bxred}{#1}} \newcommand{Bxred}[1]{\class{bxred2}{#1}} \newcommand{lrpar}[1]{\left({#1}\right)} \newcommand{lrbra}[1]{\left[{#1}\right]} \newcommand{lrabs}[1]{\left|{#1}\right|} \newcommand{bnlr}[2]{\bn{#1}\left(\bn{#2}\right)} \newcommand{nblr}[2]{\bn{#1}(\bn{#2})} \newcommand{real}[1]{\Ree\{{#1}\}} \newcommand{Real}[1]{\Ree\left\{{#1}\right\}} \newcommand{abss}[1]{\|{#1}\|} \newcommand{umin}[1]{\underset{{#1}}{\t{min}}} \newcommand{umax}[1]{\underset{{#1}}{\t{max}}} \newcommand{und}[2]{\underset{{#1}}{{#2}}} \) \( \newcommand{bn}[1]{\boldsymbol{\mathrm{#1}}} \newcommand{bns}[2]{\bn{#1}_{\t{#2}}} \newcommand{b}[1]{\boldsymbol{#1}} \newcommand{bb}[1]{[\bn{#1}]} \) \( \newcommand{abs}[1]{\left|{#1}\right|} \newcommand{ra}[]{\rightarrow} \newcommand{Ra}[]{\Rightarrow} \newcommand{Lra}[]{\Leftrightarrow} \newcommand{rai}[]{\rightarrow\infty} \newcommand{ub}[2]{\underbrace{{#1}}_{#2}} \newcommand{ob}[2]{\overbrace{{#1}}^{#2}} \newcommand{lfrac}[2]{\large{\frac{#1}{#2}}\normalsize{}} \newcommand{sfrac}[2]{\small{\frac{#1}{#2}}\normalsize{}} \newcommand{Cos}[1]{\cos{\left({#1}\right)}} \newcommand{Sin}[1]{\sin{\left({#1}\right)}} \newcommand{Frac}[2]{\left({\frac{#1}{#2}}\right)} \newcommand{LFrac}[2]{\large{{\left({\frac{#1}{#2}}\right)}}\normalsize{}} \newcommand{Sinf}[2]{\sin{\left(\frac{#1}{#2}\right)}} \newcommand{Cosf}[2]{\cos{\left(\frac{#1}{#2}\right)}} \newcommand{atan}[1]{\tan^{-1}({#1})} \newcommand{Atan}[1]{\tan^{-1}\left({#1}\right)} \newcommand{intlim}[2]{\int\limits_{#1}^{#2}} \newcommand{lmt}[2]{\lim_{{#1}\rightarrow{#2}}} \newcommand{ilim}[1]{\lim_{{#1}\rightarrow\infty}} \newcommand{zlim}[1]{\lim_{{#1}\rightarrow 0}} \newcommand{Pr}[]{\t{Pr}} \newcommand{prop}[]{\propto} \newcommand{ln}[1]{\t{ln}({#1})} \newcommand{Ln}[1]{\t{ln}\left({#1}\right)} \newcommand{min}[2]{\t{min}({#1},{#2})} \newcommand{Min}[2]{\t{min}\left({#1},{#2}\right)} \newcommand{max}[2]{\t{max}({#1},{#2})} \newcommand{Max}[2]{\t{max}\left({#1},{#2}\right)} \newcommand{pfrac}[2]{\frac{\partial{#1}}{\partial{#2}}} \newcommand{pd}[]{\partial} \newcommand{zisum}[1]{\sum_{{#1}=0}^{\infty}} \newcommand{iisum}[1]{\sum_{{#1}=-\infty}^{\infty}} \newcommand{var}[1]{\t{var}({#1})} \newcommand{exp}[1]{\t{exp}\left({#1}\right)} \newcommand{mtx}[2]{\left[\begin{matrix}{#1}\\{#2}\end{matrix}\right]} \newcommand{nmtx}[2]{\begin{matrix}{#1}\\{#2}\end{matrix}} \newcommand{nmttx}[3]{\begin{matrix}\begin{align} {#1}& \\ {#2}& \\ {#3}& \\ \end{align}\end{matrix}} \newcommand{amttx}[3]{\begin{matrix} {#1} \\ {#2} \\ {#3} \\ \end{matrix}} \newcommand{nmtttx}[4]{\begin{matrix}{#1}\\{#2}\\{#3}\\{#4}\end{matrix}} \newcommand{mtxx}[4]{\left[\begin{matrix}\begin{align}&{#1}&\hspace{-20px}{#2}\\&{#3}&\hspace{-20px}{#4}\end{align}\end{matrix}\right]} \newcommand{mtxxx}[9]{\begin{matrix}\begin{align} &{#1}&\hspace{-20px}{#2}&&\hspace{-20px}{#3}\\ &{#4}&\hspace{-20px}{#5}&&\hspace{-20px}{#6}\\ &{#7}&\hspace{-20px}{#8}&&\hspace{-20px}{#9} \end{align}\end{matrix}} \newcommand{amtxxx}[9]{ \amttx{#1}{#4}{#7}\hspace{10px} \amttx{#2}{#5}{#8}\hspace{10px} \amttx{#3}{#6}{#9}} \) \( \newcommand{ph}[1]{\phantom{#1}} \newcommand{vph}[1]{\vphantom{#1}} \newcommand{mtxxxx}[8]{\begin{matrix}\begin{align} & {#1}&\hspace{-17px}{#2} &&\hspace{-20px}{#3} &&\hspace{-20px}{#4} \\ & {#5}&\hspace{-17px}{#6} &&\hspace{-20px}{#7} &&\hspace{-20px}{#8} \\ \mtxxxxCont} \newcommand{\mtxxxxCont}[8]{ & {#1}&\hspace{-17px}{#2} &&\hspace{-20px}{#3} &&\hspace{-20px}{#4}\\ & {#5}&\hspace{-17px}{#6} &&\hspace{-20px}{#7} &&\hspace{-20px}{#8} \end{align}\end{matrix}} \newcommand{mtXxxx}[4]{\begin{matrix}{#1}\\{#2}\\{#3}\\{#4}\end{matrix}} \newcommand{cov}[1]{\t{cov}({#1})} \newcommand{Cov}[1]{\t{cov}\left({#1}\right)} \newcommand{var}[1]{\t{var}({#1})} \newcommand{Var}[1]{\t{var}\left({#1}\right)} \newcommand{pnint}[]{\int_{-\infty}^{\infty}} \newcommand{floor}[1]{\left\lfloor {#1} \right\rfloor} \) \( \newcommand{adeg}[1]{\angle{({#1}^{\t{o}})}} \newcommand{Ree}[]{\mathcal{Re}} \newcommand{Im}[]{\mathcal{Im}} \newcommand{deg}[1]{{#1}^{\t{o}}} \newcommand{adegg}[1]{\angle{{#1}^{\t{o}}}} \newcommand{ang}[1]{\angle{\left({#1}\right)}} \newcommand{bkt}[1]{\langle{#1}\rangle} \) \( \newcommand{\hs}[1]{\hspace{#1}} \)

  UNDER CONSTRUCTION

Electrostatics:
Solved Problems




\(\bb{Pr1}\)[Magnetization flux density]
A ferromagnetic sphere is uniformly magnetized. Determine the magnetic flux density at the center of the sphere.
[Sol] Key Relations:
\(\ds \bb{E1,2,3}\quad \boxed{\bvec{B}=\frac{\mu_0 I}{4\pi}\oint_{C'}\frac{d\bvec{l'}\times \uvec{R}}{R^2}}\) \(\hspace{15px}\boxed{\bvec{J}_m=\b{\nabla}\times\bvec{M}}\)\(\hspace{15px}\boxed{\bvec{J}_{ms}=\bvec{M}\times\uvec{n}}\)
  The quantity of interest is the B-field at the center of the sphere the information provided describes a field quantity – namely, magnetization. The two
  can be related via current – by first finding the total current in the region of interest (the sphere), and then the resultant B-field. \(\vplup\)
  \(\vplup\)Let \(b=\) sphere radius, \(M_0=\) magnetization intensity:
  The total current in a ferromagnetic sphere will be the combination of its surface and volume currents – which are related to the sphere’s
  magnetization via [E2,3]:
\(\bvec{J}_m=\b{\nabla}\times (M_0\uvec{z})=\boxed{0=\bvec{J}_m}\)
\(\bvec{J}_{ms}=(M_0\uvec{z})\times\underbrace{\bvec{R}}_{\uvec{n}} =\boxed{\uvec{\phi}[M_0\sin{\theta}]=\bvec{J}_{ms}}\)
  Hence, all current is on the surface of the sphere. The surface magnetization density found, \(\bvec{J}_ms\), forms a cylindrical source symmetry about the
  \(z\)-axis; thus, \(\bb{E1}\) applies. To simplify the computation, divide up the total sphere current into infinitesimal loop current elements, and sum their
  contributions to attain the total B-field.
  \(\vplup\)Refer to the figure below; the B-field due to a loop of radius \(r_0\) with (const.) current \(I\) at a point on its central axis can be determined by via \(\bb{1}\); first
  determine \(d\bvec{B}\), then integrate over \(C'\):

\(d\bvec{l'}=[r'd\phi']\uvec{\phi}=\boxed{[r_0d\phi']\uvec{\phi}=d\bvec{l'}}\)
\(\bvec{R}=\bvec{R}_{\t{field}}-\bvec{R}_{\t{source}}=\uvec{r}[0-r']+\uvec{z}[z-0]=\boxed{\uvec{z}[-r_0]+\uvec{z}[z]=\bvec{R}\vphantom{\sqrt{r_0^2+z^2}}} \Rightarrow\boxed{R=\sqrt{r_0^2+z^2}}\)
\(\ds d\bvec{l'}\times\bvec{R}=\uvec{\phi}[r_0 d\phi']\times(\uvec{r}[-r_0]+\uvec{z}[z])=[-r_0^2 d\phi']\overbrace{(\uvec{\phi}\times\uvec{r})}^{=-\uvec{z}}+[r_0 zd\phi'] \overbrace{(\uvec{\phi}\times\uvec{z})}^{=\uvec{r}}\Rightarrow\)
\(\boxed{d\bvec{l'}\times\bvec{R}=\uvec{z}[r_0^2 d\phi']+\uvec{r}[r_0 zd\phi']}\)
\(\ds\Rightarrow \frac{d\bvec{l'}\times\uvec{R}}{R^2}=\frac{d\bvec{l'}\times\bvec{R}}{R^3}=\boxed{\frac{\uvec{z}[r_0^2 d\phi']+\uvec{r}[r_0 zd\phi']}{(r_0^2+z^2)^{3/2}} =\frac{d\bvec{l'}\times\uvec{R}}{R^2}}\)

  Putting together, and recognizing that \(\uvec{r}\) varies with \(\phi'\) - that is,
\(\uvec{r}=\uvec{r}(\phi')=\uvec{x}[\cos{\phi'}]+\uvec{y}[\sin{\phi'}]\t{ - yields}\)
\(\ds\boxed{d\bvec{B}=\lrbra{\frac{\mu_0 Ir_0}{4\pi (r_0^2+z_0^2)^{3/2}}}^{\swarrow \equiv K}\hspace{-35px}(\uvec{z}[r_0]+ \uvec{x}[z\cos{\phi'}]+\uvec{y}[z\sin{\phi'}])d\phi'}\)

  The path of integration is along the source: \(C':\underbrace{r'=r_0}_{=\t{const}},0\leq\phi'\leq2\pi,\underbrace{z'=0}_{=\t{const}}\). Integrating,
\(\ds\begin{align}\bvec{B}&=\oint_{C'}d\bvec{B}=K\left[\vphantom{\intlim{0}{2\pi}}\right. \uvec{z}r_0\underbrace{\intlim{0}{2\pi}d\phi'}_{=2\pi}+\uvec{x}z\underbrace{\intlim{0}{2\pi}\cos{\phi'}d\phi'}_{=0:\ [\sin{\phi'}] _0^{2\pi}=0}+\uvec{y}z\underbrace{\intlim{0}{2\pi}\sin{\phi'}d\phi'}_{=0:\ [-\cos{\phi'}]_0^{2\pi}=0}\left.\vphantom{\intlim{0}{2\pi}}\right]\\ &= \uvec{z}[2\pi r_0K]=\uvec{z}\left[(2\pi r_0)\Frac{\mu_0 Ir_0}{4\pi(r_0^2+z^2)^{3/2}}\right]\Rightarrow\end{align}\)
\(\boxed{\bvec{B}_{\t{loop}}=\uvec{z}\left[\frac{1}{2}\mu_0 r_0^2I\right]\frac{1}{(r_0^2+z^2)^{3/2}}}\ \ \bb{1^*}\)
  Cylindrical source symmetry could have also been used to justify a claim that field cylindrical \(\uvec{r}\)-components cancel to zero.
  \(\vplup\)For the sphere, the total B-field at the origin can be determined via the vector-sum of differential field contributions from differential current source
  loops distributed over the sphere’s surface:
  The surface integral of \(d\bvec{B}_{\t{loop}}\) contributions will yield the total magnetic flux intensity, \(\bvec{B}\), at the center of the sphere. To express the line current
  element \(dI_m\) in terms of the surface current density \(\bvec{J}_{\t{ms}}\), apply a virtual impulse construct – as no ‘real’ surface is oriented orthogonal to the
  spherical shell of interest:
  The R-limits of integration aren’t taken to be \((-\infty,\infty)\), as that would include the contribution of a separate loop, positioned symmetrically opposite
  about the origin to the one of interest. Defining \(I_m\) to be the total current resultant from integrating \(\bvec{J}_m\) over the surface of the sphere, we get:
\(\ds I_m=\iint_S\bvec{J}_{\t{m}}\cdot d\bvec{S}\)
  A virtual impulse construct can be applied here by expressing the volume current density \(\bvec{J}_{\t{m}}\) in terms of the surface current density \(\bvec{J}_{\t{ms}}\) and an
  impulse function: \(\bvec{J}_{\t{m}}=\uvec{\phi}[J_{\t{ms}}\delta (R-b)]\). Then,
\(\begin{align}I_m &= \intlim{0}{\pi}\lrbra{\intlim{R=0}{R\rightarrow\infty}[RJ_{\t{ms}}\delta (R-b)]dR}d\theta \overbrace{[\uvec{\phi}\cdot\uvec{\phi}]}^{=1}\\ &= \intlim{0}{\pi}bJ_{\t{ms}}d\theta\Rightarrow\boxed{dI_m=J_{\t{ms}}bd\theta}\end{align}\)
  Applying the differential form of \(\bvec{B}_{\t{loop}}\) in \(\bb{1^*}\), and taking source coordinates \((\theta\rightarrow\theta')\), we get
\(\ds d\bvec{B}=\uvec{z}\lrbra{\frac{\mu_0}{2}}\frac{\overbrace{r_0^2}^{=R\sin{\theta'}}}{\underbrace{(r_0^2+z^2)^{3/2}}_{=R^3}}dI_m= \uvec{z}\lrbra{\frac{\mu_0}{2}}\frac{\t{sin}^2(\theta')}{\underbrace{R}_{=b,=\t{const}}}\overbrace{J_{\t{ms}}}^{=M_0\sin{\theta'}}bd\theta'\Rightarrow\)
\(\ds\boxed{d\bvec{B}=\underbrace{\uvec{z}\lrbra{\frac{\mu_0 M_0}{2}}}_{\equiv K'}\t{sin}^3(\theta')d\theta'}\)
  Summing the differential loop B-field contributions over the surface of the sphere, we get
\(\ds\bvec{B}=K\intlim{\theta'=0}{\theta'=\pi}\t{sin}^3(\theta')d\theta' =K\intlim{\theta'=0}{\theta'=\pi}\underbrace{\sin{\theta'} (1-\t{cos}^2(\theta'))d\theta'}_{u=\cos{\theta'},\ du=-\sin{\theta'}d\theta' \\ =\int (u^2-1)du=u^3/3-u} = \frac{K}{3}\overbrace{[\t{cos}^3(\theta')-3 \cos{\theta'}]_0^\pi}^{=4}\Rightarrow\)\[\bxred{\bvec{B}=\uvec{z}\lrbra{\frac{2}{3}\mu_0 M_0}}\]


Dragon Notes,   Est. 2018     About

By OverLordGoldDragon