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?

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