Circuits and electricity

What is Kirchhoff’s voltage law (KVL)?

$$\oint_{C}\underline{E}\cdot\underline{dl}=0$$

Where E is the electrostatic field and dl is a vector element of the path/loop.

The sum of all electromotive forces and potential differences around any closed loop of a circuit is zero.

Equation for vector current density

$$\underline{j}=nq\underline{v}_{d}$$

where n is the concentration of moving charged particles, q is charge per particle and v_d is drift velocity.

(Current per cross-sectional area - scalar)

Current density

$$dI=\underline{j}\cdot\underline{dS}$$

What is Kirchhoff’s current law (KCL)?

$$\frac{\partial\rho_{c}}{\partial t}+\underline{\nabla}\cdot\underline{j}=0$$

Where j is the current density vector.

The sum of all currents entering a junction equals the sum of all currents leaving a junction.

Draw receptor convention (circuits)

Draw generator convention (circuits)

What is the electromotive force?

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What is Faraday’s Law?

$$\oint_{C}\underline{E}\cdot d\underline{l}=-\frac{d\Phi_{B}}{dt}$$

What is Lenz’s law?

The direction of magnetic induction effect is in the direction opposing the cause of that effect.

Maxwell-flux equation

$$\underline{\nabla}\cdot\underline{B}=0$$

Maxwell-Gauss equation

$$\underline{\nabla}\cdot\underline{E}=\frac{\rho_{c}}{\epsilon_0}$$

Maxwell-Faraday equation

$$\underline{\nabla}\times\underline{E}=-\frac{\partial{\underline{B}}}{\partial t}$$

Maxwell-Ampere equation

$$\underline{\nabla}\times\underline{B}=\mu_0\underline{j}+\mu_0\epsilon_0\frac{\partial\underline{E}}{\partial t}$$

Laplace force (derived from magnetic of Lorenz)

$$\underline{F_{L}}=\int_{wire}I\underline{dl}\times\underline{B}$$

Resistance equation (receptor convention)

$$V_{R}=RI$$

Capacitance equation (receptor convention)

$$V_{C}=\frac{Q}{C}$$

Equation for capacitor in parallel and in series (receptor convention)

• In parallel: $C_{eq}=C_1+C_2$

• In series: $\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}$ or $C_{eq}=\frac{C_1C_2}{C_1+C_2}$

Voltage of an inductor equation (receptor convention)

$V_{L}=L\frac{dI}{dt}$ where L is inductance

Power consumed by an electrical component (receptor convention)

$$P=VI$$

What is the electric field inside a conductor?

$$\underline{E}=0$$

Current and current density equation

$$I=\iint_{S}\underline{j}\cdot d\underline{S}$$

Definition of self-inductance

$\Phi_{B}=LI$ where L is the self-inductance

The ability of a circuit to create magnetic flux through itself

Definition of mutual inductance

$\Phi_{B}=MI$ where M is the mutual inductance (between two circuits)

The ability of a circuit to create magnetic flux through another circuit is mutual inductance.

Known result for self-inductance of a solenoid

$$L=\mu_0n^2Sl$$

How do we find mutual inductance (M)?

• Calculate magnetic field through one circuit

• Calculate the flux through the second circuit

• Use the definition of mutual inductance to find M

Mutual inductance of two solenoids (known result)

$$M=\mu_0n_1n_2Sl$$

What is the surface enclosed by a solenoid?

$S\cdot n_{}$ where n is the number of loops. And S is approximated to the area of a circle.

Magnetic curl definition

• A measure of whether the electromagnetic field rotates around a point and how much

• essential for determining how electric currents and changing electric fields generate rotating magnetic fields

• $$\underline{curl}\left(\underline{A}\right)=\underline{\nabla}\times\underline{A}$$

• curl >0 is anti-clockwise, < 0 is clockwise and =0 is no curl

Power supplied by a voltage source

$$P_{}=I_{cell}\cdot V_{ter\min al}$$