Study guide: Electromagnetism and Charge Conservation

📋 Course Outline

1. Local Maxwell Equations
2. Surface and Line Charge
3. Electric Current Density
4. Charge Conservation
5. Stationary Current and Nodes
6. Integral Gauss and Ampère Laws
7. Magnetic Flux and Faraday’s Law
8. Electric Field at a Surface

📖 1. Local Maxwell Equations

★ Must-know

📐 Formula - Maxwell-Gauss’ local equation is $\operatorname{div}(\vec E)=\frac{\rho(M,t)}{\varepsilon_0}$.

📐 Formula - Maxwell-Faraday’s local equation is $\operatorname{rot}(\vec E)=-\frac{\partial \vec B}{\partial t}$.

📐 Formula - The local magnetic-flux equation is $\operatorname{div}(\vec B)=0$.

📐 Formula - Maxwell-Ampère’s local equation is $\operatorname{rot}(\vec B)=\mu_0\vec j+\mu_0\varepsilon_0\frac{\partial \vec E}{\partial t}$.

Further detail

• The vacuum electric permittivity is approximately $8.85\times10^{-12}\ \mathrm{F\cdot m^{-1}}$.

📐 Formula - The vacuum constants satisfy $\mu_0\varepsilon_0=\frac{1}{c^2}$, where $c$ is the speed of electromagnetic waves in vacuum.

📖 2. Surface and Line Charge

🔑 Key Concepts & Definitions

• Surface charge density: Surface charge density is the charge per area, $\sigma(M)=\frac{dq(M)}{dS(M)}$.
• Linear charge density: Linear charge density is the charge per length, $\lambda=\frac{dq(M)}{d\ell(M)}$.

📝 Essential Points

📐 Formula - The total charge on a surface distribution is $Q_S=\iint_{M\in(S)}\sigma(M)\,dS$.

• The electric field $\vec E$ is not defined under the surface and line charge distributions described here.

📖 3. Electric Current Density

★ Must-know

📐 Formula - For one type of charge carrier, the current density is $\vec j_1(P)=n_{1m}q_{1m}\vec v_1(P)$.

📐 Formula - The current through a surface due to one carrier type is $i_{S1}=\iint_{P\in(S)}\vec j_1(P)\cdot d\vec S$.

📐 Formula - The total current through a surface is $i_{(S)}=\iint_{P\in(S)}\vec j(P)\cdot d\vec S(P)$.

Further detail

📐 Formula - For multiple carrier types, the total current density is $\vec j=\sum_k n_{km}q_{km}\vec v_k(P)$.

• Current density $\vec j$ is measured in $\mathrm{A\cdot m^{-2}}$.

📖 4. Charge Conservation

★ Must-know

📐 Formula - In one dimension, charge conservation is $\frac{\partial\rho}{\partial t}+\frac{\partial j}{\partial x}=0$.

📐 Formula - In three dimensions, local charge conservation is $\frac{\partial\rho}{\partial t}+\operatorname{div}(\vec j)=0$.

Further detail

📐 Formula - The charge inside a fixed volume is $Q_{(V)}(t)=\iiint_{M\in(V)}\rho(M,t)\,d\tau(M)$.

📐 Formula - The current crossing a closed surface is $i_S(t)=\oiint_{P\in(S)}\vec j(P,t)\cdot d\vec S(P)$.

• When the current density points along the outward surface element, charge leaves the volume and the volume loses charge.

💡 Memory Hook

Charge leaving a volume decreases its contained charge.

📖 5. Stationary Current and Nodes

★ Must-know

📐 Formula - In a stationary regime, $\frac{\partial\rho}{\partial t}=0$ and therefore $\operatorname{div}(\vec j)=0$.

📌 In a stationary regime, the total current through any closed surface is zero.

Further detail

📐 Formula - At a junction in a stationary regime, the signed currents satisfy $I_1+I_2+I_3=0$.

📌 For a tube of current density with no current through its lateral surface, the current entering one end equals the current leaving the other, $I_1=I_2$.

📖 6. Integral Gauss and Ampère Laws

★ Must-know

📐 Formula - The integral Gauss law is $\oiint_{P\in(S)}\vec E(P)\cdot d\vec S(P)=\frac{Q_{\mathrm{int}}(V)}{\varepsilon_0}$.

📐 Formula - The generalized integral Ampère law is $\oint_{P\in(C)}\vec B(P,t)\cdot d\vec\ell(P)=\mu_0 I(S)+\mu_0\varepsilon_0\iint_{M\in(S)}\frac{\partial\vec E}{\partial t}\cdot d\vec S(M)$.

Further detail

• Electric charges are sources of the electric field $\vec E$.

📐 Formula - In a stationary regime, Ampère’s law is $\oint_{P\in(C)}\vec B(P,t)\cdot d\vec\ell(P)=\mu_0 I$.

• The contour direction in the Ampère integral is set by the right-hand rule relative to the oriented surface.

💡 Memory Hook

Enclosed charge sources electric flux; moving charges source magnetic circulation.

📖 7. Magnetic Flux and Faraday’s Law

★ Must-know

📐 Formula - The magnetic flux through a surface is $\Phi=\iint_{M\in(S)}\vec B(M,t)\cdot d\vec S(M)$.

📐 Formula - For an immobile surface, the Maxwell-Faraday integral law is $\oint_{P\in(C)}\vec E(P,t)\cdot d\vec\ell(P)=-\frac{d\Phi}{dt}$.

• The electromotive force is $e=-\frac{d\Phi}{dt}$.

Further detail

• The magnetic field $\vec B$ has zero net flux through a closed surface, $\oiint_{P\in(S)}\vec B(P)\cdot d\vec S(P)=0$.

• The four coupled equations in $\vec E$ and $\vec B$ are useful for describing electromagnetic waves.

💡 Memory Hook

Changing magnetic flux produces an electromotive force.

📖 8. Electric Field at a Surface

📝 Essential Points

📐 Formula - For the surface-charge case shown, the field relation is $\vec E(P_1)-\vec E(P_2)=\frac{\sigma}{\varepsilon_0}$.