E+M Key Equations

Gauss’ Law for electric fields (M1)

$$\int E\cdot dA=\frac{Q_{enc}}{\epsilon_0}$$

Gauss’ Law for magnetic fields (M2)

$$\int B\cdot dA=0$$

Ampere’s Law

$$\int B\cdot dl=\mu_0I_{enc}$$

Maxwell’s correction to Ampere’s Law (M3)

\int B\cdot dl=\mu_0\int\left(\imaginaryJ+\epsilon_0\frac{\partial E}{\partial t}\right)\cdot dA

Where $j$ is the current density. (not the unit vector)

Faraday’s Law of Induction (M4)

$$\int E\cdot dl=-\frac{d}{dt}\int_{S}B\cdot dA$$

Coulumb’s Law

$$F=\frac{q_1q_2}{4\pi\epsilon_0r^2}\underline{r}$$

Key Gauss Law solutions: Point charge

$$E=\frac{q}{4\pi\epsilon_0r^2}\underline{r}$$

Key Gauss Law solutions: Line charge

$$E=\frac{\lambda}{2\pi r\epsilon_0}\underline{r}$$

$\lambda$ is charge density (per unit distance)

Key Gauss Law solutions: Plane of charge (including parallel plate capacitors)

$$E=\frac{\sigma}{2\epsilon_0}\underline{r}$$

$\sigma$ is charge density (per unit area)

Scalar electric potential

$$\phi\left(r\right)=-\int^{r}_{\infty}E\cdot dl$$

Electric field from potential

$E=-\nabla\phi=-\frac{d\phi}{dx}$ (if 1D)

Equation of current from current density (recall both current density and area are vectors…)

$$I=\int_{s}j\cdot dA$$

Electric field energy density

$u_{E}=\frac{\epsilon_0E^2}{2}$
True for all electric fields, easy to solve for parallel plate capacitor.

To get electric field potential energy, just multiply by volume (i.e for parallel plate capacitor example will be area of one plate times distance to other plate)

Total energy stored in a capacitor

$$U=\frac12CV^2$$

Recall $C=\frac{Q}{V}$

Electric dipole moment and torque on electric dipole

$$p=qd$$

$p$ is the electric diplole moment, note both this and distance are vectors

$$\tau=p\times E$$

Electric dipole potential

$$\phi\left(\theta\right)=\frac{Qd\cos\theta}{4\pi\epsilon_0r^2}$$

Electric dipole electric field

$$E=\frac{Qd}{4\pi\epsilon_0}\left(2\cos\theta\underline{r}+\sin\theta\underline{\theta}\right)$$

Electric dipole energy

$$U=-p\cdot E$$

Magnetic force (on moving charged thing)

$$F=q\left(v\times B\right)$$

note this rule works for all equations $c=a\times b$

Lorentz force

$$F=q\left(E+v\times B\right)$$

simply combination of magnetic force ($F=q\left(v\times B\right)$) and electric force ($F=qE$)

Magnetic force on a charged wire

$$F=l\left(I\times B\right)$$

Equation relating electric current to individual charge carriers

$$I=nAqv$$

where $n$ is the charge carrier density (number of charge carriers per unit length). Note that this is how the magnetic force on a current carrying wire equation is derived (put this into force on moving charge equation)

The Biot-Savart Law

$$dB=\frac{\mu_0I}{4\pi r^2}dl\times\underline{r}$$

note that as $\underline{r}$ is a unit vector, the magnitude of this equation simplifies to $\left\vert dB\right\vert=\frac{\mu_0I}{4\pi r^2}dl$

Key Biot-Savart Law solutions: Magnetic field from loop of current

At centre:

$$B\left(z=0\right)=\frac{\mu_{o}I}{2a}\underline{z}$$

Just z component at any z value:

$$B_{z}=\frac{\mu_{o}Ia^2}{2\left(a^2+z^2\right)^{\frac32}}\underline{z}$$

Key Biot-Savart Law solutions: Magnetic field from an infinite wire

$$B\left(s\right)=\frac{\mu_0I}{2\pi s}\underline{\theta}$$

Force experienced by parallel currents

$$\frac{F}{l}=\frac{\mu_0I_1I_2}{2\pi s}$$

Magnetic flux

$\Phi=\int_{s}B\cdot dA$
Note, different from Gauss’ Law for magnetic fields as that law is the complete circulation over a gaussian surface, whereas this is an integral over part of a given surface

Induced EMF by a magnetic field

Expressed by Faraday’s law, but this time in terms of EMF:

\char"0190 =-\int E\cdot dl=-\frac{d\Phi}{\differentialD t}

Recall magnetic flux defined as $\Phi=\int_{s}B\cdot dA$. This shows where the version of Faraday’s law used as M4 comes from.

Magnetic field produced in a solenoid

$$B_{in}=\mu_0nI$$

Note that $B_{in}$ is specifically magnetic field inside solenoid (= 0 outside… kind of)

$n$ is coil density (number of coils per unit length)

Moment on magnetic dipole

$$\mu=IA$$

Torque on magnetic dipole

$$\tau=\mu\times B$$

Magnetic dipole energy

$$U=-\mu\cdot B$$

magnetic field energy density

$$u_{B}=\frac{B^2}{2\mu_0}$$

Multiply by volume to find magnetic potential energy

Self inductance and back voltage

$$\Phi=LI$$

note that from equation for EMF we see that \char"0190 =-\frac{d\Phi}{\differentialD t}=-L\frac{dI}{\differentialD t}

Back voltage defined as negative of this EMF: $\phi_{L}=-\char"0190$

Self inductance of a solenoid

$$L=\mu_0n^2lA$$

$n$ is number of coils per unit distance, and $l$ is the total length of the coil

Total energy stored in an inductor

$$U=\frac12LI^2$$

Mutual inductance (between two solenoids)

$$\Phi_{21}=M_{21}I_1$$

$I_1$ is the current in the first coil (which induces an emf in the second one), $\Phi_{12}$ is the flux linkage between the coils (magnetic flux in both coils), and $M_{21}$ is the mutual inductance - the coupling coefficient between coils, which can also be written as:

$$M_{21}=\frac{\mu_0N_1N_2A}{l}$$

$N_1$ and $N_2$ is the total number of coils in each solenoid

Transformer EMF

$$\char"0190 _1=\frac{N_1}{N_2}\char"0190 _2$$

$N_1$ and $N_2$ is the total number of coils in each solenoid

Transformer power

Power in transformers is conserved:

$$\char"0190 _1I_1=\char"0190 _2I_2$$