Understanding Electric Fields, Representations, and Charge Distributions (AP Physics 2 Unit 2)

Electric field

A description of how source charges influence space; any charge placed in the field experiences an electric force.

Test charge

A small, imaginary positive charge used to define and probe the electric field at a point; the field is due to source charges, not the test charge.

Electric field definition (vector)

(\vec{E}=\frac{\vec{F}}{q}): the force per unit positive test charge at a location; direction is the direction a positive charge would be pushed.

Force from an electric field

(\vec{F}=q\vec{E}): the force on any charge placed in a known electric field.

Electric field units

Common units are newtons per coulomb (N/C); equivalent units are volts per meter (V/m).

Coulomb constant (k)

(k\approx 8.99\times10^9\ \text{N·m}^2/\text{C}^2); appears in point-charge electric field and force formulas.

Vector field

A field that has both magnitude and direction at every point in space; (\vec{E}) must be treated as a vector, not a scalar.

Point-charge electric field magnitude

(E=\frac{k|q|}{r^2}): the field strength from a single point charge decreases with the square of the distance.

Point-charge electric field (vector form)

(\vec{E}=\frac{kq}{r^2}\hat{r}): (\hat{r}) points from the source charge to the field point; the sign of (q) sets inward vs outward direction.

Inverse-square law (electric field)

The field from a point charge falls off as (1/r^2) because the influence spreads over the surface area of an expanding sphere.

Radial field pattern

For a point charge, field lines point radially outward for positive charge and radially inward for negative charge.

Superposition principle (electric fields)

Net electric field is the vector sum of contributions from all sources: (\vec{E}{\text{net}}=\sumi \vec{E}_i).

Component addition (for fields)

Because (\vec{E}) is a vector, you add x- and y-components (not magnitudes) when combining fields from multiple charges.

Field direction convention

The direction of (\vec{E}) is defined by the force on a positive test charge; a negative charge feels a force opposite (\vec{E}).

Electric field lines

A drawing tool that represents the electric field: tangent gives direction of (\vec{E}), and line density indicates relative magnitude.

Field line density

Qualitative indicator of field strength: closer (denser) field lines mean a larger magnitude of (E).

Field line start/end rule

In electrostatics, field lines start on positive charge and end on negative charge (or at infinity if no opposite charge is available).

Field lines cannot cross

Crossing would imply two different directions for (\vec{E}) at the same point, which is impossible since (\vec{E}) is single-valued at each location.

Conductor

A material in which charges (typically electrons) can move freely through the material.

Insulator

A material in which charges are bound and do not move freely through the object.

Electrostatic equilibrium (conductor)

A state where charges in a conductor are no longer moving; implies specific rules for charge location and electric field behavior.

Conductor in equilibrium: key properties

(1) Net electric field inside the conducting material is zero, (2) excess charge resides on the surface, (3) field just outside is perpendicular to the surface.

Charge densities

Ways to describe distributed charge: linear (\lambda=Q/L), surface (\sigma=Q/A), and volume (\rho=Q/V).

Spherical symmetry (acts like a point charge)

Outside a uniformly charged spherical shell or charged conducting sphere, the field behaves as if all charge were at the center: (E=\frac{k|Q|}{r^2}) for (r) outside the sphere.

Electrostatic induction

Charge rearrangement in a conductor caused by a nearby charged object; can produce attraction and field patterns even if the conductor remains net neutral.