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Exam, electromagnetism - Coggle Diagram
Exam, electromagnetism
Chapter 1
1.1.4
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\(\boldsymbol{\eta}\) er separasjonsvektoren, \(\boldsymbol{r}\) er feltpunktet, dvs der man regner ut det elektriske/magnetiske feltet, og\(\boldsymbol{r'}\) er ladningspunktet.
1.2.3
vektoroperator \(\boldsymbol{\nabla}=\boldsymbol{\hat{x}}\frac{\partial}{\partial x}+\boldsymbol{\hat{y}}\frac{\partial}{\partial y}+\boldsymbol{\hat{z}}\frac{\partial}{\partial z}\)
\(\boldsymbol{\nabla f}=\left(\boldsymbol{\hat{x}}\frac{\partial}{\partial x}+\boldsymbol{\hat{y}}\frac{\partial}{\partial y}+\boldsymbol{\hat{z}}\frac{\partial}{\partial z}\right)\boldsymbol{f}=\boldsymbol{\hat{x}}\frac{\partial\boldsymbol{f}}{\partial x}+\boldsymbol{\hat{y}}\frac{\partial\boldsymbol{f}}{\partial y}+\boldsymbol{\hat{z}}\frac{\partial\boldsymbol{f}}{\partial z}\)
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1.3.3
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\(\int_a^b\nabla f\cdot dI=\int_a^b\,df=f(b)-f(a)\)
1.3.4
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\(\int_V(\nabla\cdot v)\,d\tau=\oint_Sv\cdot da\)
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1.2.2
\(df=\left(\frac{\partial f}{\partial x}\right)dx+\left(\frac{\partial f}{\partial y}\right)dy+\left(\frac{\partial f}{\partial z}\right)dz\)
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\(df=\left(\frac{\partial f}{\partial x}\hat{x}+\frac{\partial f}{\partial y}\hat{y}+\frac{\partial f}{\partial z}\hat{z}\right)\cdot\left(dx\hat{x}+dy\hat{y}+dz\hat{z}\right)=\nabla f\cdot dI\)
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Chapter 2
2.1.3
\(\boldsymbol{F}=\boldsymbol{QE}(\boldsymbol{r})=\boldsymbol{Q}\frac{1}{4\pi\varepsilon_0}\sum\frac{q_i}{\eta_i^2}\boldsymbol{\hat{\eta}_i}\)
2.1.4
Line charge: \(\boldsymbol{E}(\boldsymbol{r})=\frac{1}{4\pi\varepsilon_0}\int\frac{\lambda(\boldsymbol{r'})dI'}{\eta^2}\boldsymbol{\hat{\eta}}\)
Surface charge: \(\boldsymbol{E}(\boldsymbol{r})=\frac{1}{4\pi\varepsilon_0}\int\frac{\sigma(\boldsymbol{r'})da'}{\eta^2}\boldsymbol{\hat{\eta}}\)
Volume charge: \(\boldsymbol{E}(\boldsymbol{r})=\frac{1}{4\pi\varepsilon_0}\int\frac{\rho(\boldsymbol{r'})d\tau'}{\eta^2}\boldsymbol{\hat{\eta}}\)
2.2.1
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Gauss law, gaussian surface
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2.2.4
static charge distribution has zero curl, because every static charge has radial electric field
\(\boldsymbol{\nabla}\times\boldsymbol{E}=\boldsymbol{\nabla}\times(\boldsymbol{E_1}+\boldsymbol{E_2}+\cdots)=(\boldsymbol{\nabla}\times\boldsymbol{E_1})+(\boldsymbol{\nabla}\times\boldsymbol{E_2})+\cdots=0\)
2.3.1
\(\boldsymbol{V}(\boldsymbol{r})=-\int_\mathcal{O}^\boldsymbol{r}\boldsymbol{E}\cdot \boldsymbol{dI}\)
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\(\boldsymbol{V}(\boldsymbol{b})-\boldsymbol{V}(\boldsymbol{a})=-\int_\boldsymbol{a}^\boldsymbol{b}\boldsymbol{E}\cdot \boldsymbol{dI}\)
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2.4.3
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Volume energy: \(W=\frac{\varepsilon_0}{2}\int E^2\,d\tau\)
Volume energy: \(W=\frac{1}{2}\int \rho V\,d\tau\)
Line energy: \(W=\frac{1}{2}\int \lambda V\,dI\)
Surface energy: \(W=\frac{1}{2}\int \sigma V\,da\)
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Chapter 3
3.1
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2D: \(V(x,y)=\frac{1}{2\pi R}\oint_\text{circle}V\,dI\)
No local minima or maxima tolerated, only endpoints maxima/minima
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3D: \(V(r)=\frac{1}{4\pi R²}\oint_\text{sphere}V\, da\)
3.2
find electric field above conducting plane and charge -> find electric field above origo with same charge and a opposite charge same distance below \(z=0\)
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Chapter 4
4.1.1
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Two classes, conductors and insulators/dielectrics
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4.2.1
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\(V(r)=\frac{1}{4\pi\varepsilon_0}\oint_S\frac{\sigma_b}{\eta}\,da'+\frac{1}{4\pi\varepsilon_0}\int_V\frac{\rho_b}{\eta}\,d\tau'\)
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Chapter 5
5.1.2
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\(F_{\text{total}}=F_{\text{magnetic}}+F_{\text{electric}}=Q[\boldsymbol{E}+(\boldsymbol{v}\times\boldsymbol{B})]\)
5.1.3
Magnetic force on line current \(F=\int(\boldsymbol{v}\times\boldsymbol{B})\,\lambda\,dI=\int(\boldsymbol{I}\times\boldsymbol{B})\,dI=I\int(d\boldsymbol{I}\times\boldsymbol{B})\)
Surface current density \(\boldsymbol{K}=\sigma\boldsymbol{v}=\frac{\text{current}}{\text{perpendicular width}}\)
Magnetic force on surface current \(F=\int(\boldsymbol{v}\times\boldsymbol{B})\,\sigma\,da=\int(\boldsymbol{K}\times\boldsymbol{B})\,da\)
Volume current density \(\boldsymbol{J}=\rho\boldsymbol{v}=\frac{\text{current}}{\text{perpendicular area}}\)
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Magnetic force on volume current \(F=\int(\boldsymbol{v}\times\boldsymbol{B})\,\rho\,d\tau=\int(\boldsymbol{J}\times\boldsymbol{B})\,d\tau\)
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