Unit 1 Learning Notes: Electric Fields, Charge Distributions, Gauss’s Law, and Conductors

Electric field ($\vec{E}$)

A vector field in space defined as force per unit positive test charge: ($\vec{E}=\frac{\vec{F}}{q}$); exists whether or not a test charge is present.

Test charge

A small (ideally positive) charge used to probe an electric field without significantly disturbing the source charge configuration.

Electric field units

($\mathrm{N/C}$) (newtons per coulomb), equivalent to ($\mathrm{V/m}$) (volts per meter).

Coulomb (point-charge) electric field

Field of a stationary point charge ($Q$): ($\vec{E}(\vec{r})=\frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}\hat{r}$), radial with magnitude ($\propto \frac{1}{r^2}$).

Permittivity of free space ($\epsilon_0$)

Physical constant appearing in Coulomb’s law and Gauss’s law that sets the strength of electric interactions in vacuum.

Radial unit vector ($\hat{r}$)

A unit vector pointing outward from the source charge to the field point; the sign of ($Q$) determines whether ($\vec{E}$) points with or against ($\hat{r}$).

Superposition (electric fields)

The net electric field is the vector sum of fields from all sources: ($\vec{E}_{\text{net}}=\sum_i \vec{E}_i$) (or an integral for continuous charge).

Electric force from a field

Force on a charge ($q$) placed in an electric field: ($\vec{F}=q\vec{E}$) (direction flips if ($q<0$)).

Electric field lines

A visualization tool where the tangent gives the direction of ($\vec{E}$) and the line density indicates relative magnitude; lines start on + charge and end on − charge (or infinity).

No-crossing rule (field lines)

Electric field lines cannot cross because the electric field at a point has a unique direction.

Line charge density ($\lambda$)

Charge per unit length for a continuous distribution: ($\lambda = \frac{dq}{ds}$), units ($\mathrm{C/m}$).

Surface charge density ($\sigma$)

Charge per unit area on a surface: ($\sigma = \frac{dq}{dA}$), units ($\mathrm{C/m^2}$).

Volume charge density ($\rho$)

Charge per unit volume in a material: ($\rho = \frac{dq}{dV}$), units ($\mathrm{C/m^3}$).

Charge element ($dq$)

An infinitesimal piece of charge used in integration for continuous distributions (e.g., ($dq=\lambda\,ds$), ($dq=\sigma\,dA$), ($dq=\rho\,dV$).

Field contribution from a charge element ($d\vec{E}$)

Infinitesimal field from ($dq$): ($d\vec{E}=\frac{1}{4\pi\epsilon_0}\frac{dq}{R^2}\hat{R}$), where ($\hat{R}$) points from source element to field point.

Symmetry cancellation (in field integrals)

Using symmetry to argue certain vector components of (\vec{E}) cancel (e.g., transverse components from opposite sides of a ring).

Uniformly charged ring (on-axis field)

For a ring of radius ($a$) and total charge ($Q$), on its axis a distance ($x$) from center: ($E_x=\frac{1}{4\pi\epsilon_0}\frac{Qx}{(a^2+x^2)^{3/2}}$) along the axis.

Electric flux ($\Phi_E$)

Measure of electric field passing through a surface: ($d\Phi_E=\vec{E}\cdot d\vec{A}$); depends on angle via the dot product.

Gauss’s law (integral form)

Relates flux through a closed surface to enclosed charge: ($\oint \vec{E}\cdot d\vec{A}=\frac{Q_{\text{enc}}}{\epsilon_0}$).

Gaussian surface

An imaginary closed surface chosen to exploit symmetry so that ($\oint \vec{E}\cdot d\vec{A}$) can be evaluated easily and solved for ($E$).

Infinite line charge field (Gauss result)

For an infinite line with uniform ($\lambda$): ($E(r)=\frac{\lambda}{2\pi\epsilon_0 r}$), directed radially outward for ($\lambda > 0$).

Infinite sheet of charge field (Gauss result)

For an infinite sheet with uniform ($\sigma$): ($E=\frac{\sigma}{2\epsilon_0}$), perpendicular to the sheet and independent of distance.

Electrostatic equilibrium (conductor)

Condition in which charges in a conductor are at rest on average; implies ($\vec{E}=0$) inside the conducting material and excess charge resides on the surface.

Conductor boundary condition (normal field)

At a conductor surface in electrostatics: (E\perp^{ ext{out}}-E\perp^{ ext{in}}=\sigma/\epsilon0); since (E\perp^{ ext{in}}=0), (E\perp^{ ext{out}}=\sigma/\epsilon0).

Faraday cage (electrostatic shielding)

Shielding effect where charges rearrange on a conductor so electric fields are canceled within the conductor (and in an empty enclosed cavity) in electrostatic equilibrium.