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}=\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 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}}=\sumi \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 = dq/ds), units (\mathrm{C/m}).

Surface charge density ((\sigma))

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

Volume charge density ((\rho))

Charge per unit volume in a material: (\rho = 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: (Ex=\frac{1}{4\pi\epsilon0}\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}}}{\epsilon0}).

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^{\text{out}}-E\perp^{\text{in}}=\sigma/\epsilon0); since (E\perp^{\text{in}}=0), (E\perp^{\text{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.