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

Electric Field

What an electric field is

An electric field is a way to describe how electric charges influence the space around them. Instead of thinking “charge Q pulls on charge q with some force,” you think “charge Q creates a field everywhere, and any charge placed in that field feels a force.” This matters because the field exists whether or not a “test charge” is present—so the field is a property of the configuration of source charges.

Formally, the electric field vector \vec{E} at a point is defined as the force per unit positive test charge placed at that point:

\vec{E} = \frac{\vec{F}}{q}

  • \vec{F} is the electric force on a small test charge.
  • q is the (positive) test charge.
  • Units: \mathrm{N/C} (newtons per coulomb), which is equivalent to \mathrm{V/m}.

A key idea hidden in this definition: \vec{E} is defined at a point in space, not “on a charge.” Forces act on charges; fields exist in space.

Why fields are useful

Fields let you:

  • Predict forces on any charge without re-deriving Coulomb’s law each time.
  • Use superposition cleanly: fields from multiple sources add as vectors.
  • Connect electricity to energy through electric potential V. In electrostatics, fields and potentials are two complementary ways to describe the same physics.

Electric field of a point charge (Coulomb field)

A stationary point charge Q produces a spherically symmetric field. Using Coulomb’s law and the definition of field:

\vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}\hat{r}

  • \epsilon_0 is the permittivity of free space.
  • r is the distance from the charge to the observation point.
  • \hat{r} points radially outward from the charge.

Direction matters:

  • If Q>0, \vec{E} points away from the charge.
  • If Q

A common misconception is to treat \hat{r} as “always outward.” It’s outward from the source charge, but the sign of Q determines the actual field direction.

Superposition of electric fields

Electric fields obey linear superposition: the net field is the vector sum of contributions from all source charges.

For discrete charges:

\vec{E}_{\text{net}} = \sum_i \vec{E}_i

This is easy to misuse if you add magnitudes instead of vectors. When symmetry is not perfect, you must resolve components.

Field lines (a visualization tool)

Electric field lines are a picture, not a physical object.

  • The tangent to a field line gives the direction of \vec{E}.
  • The density of lines represents the relative magnitude of \vec{E}.
  • Lines begin on positive charge and end on negative charge (or at infinity).

Mistake to avoid: field lines cannot cross. If they crossed, the field would have two directions at a point.

Worked example: field and force at a point

A charge Q=+3.0\,\mu\mathrm{C} is at the origin. Find \vec{E} and the force on a test charge q=-2.0\,\mu\mathrm{C} located at r=0.50\,\mathrm{m} on the positive x axis.

1) Electric field magnitude at the location:

E = \frac{1}{4\pi\epsilon_0}\frac{|Q|}{r^2}

Direction: since Q>0 and the point is on +x, \vec{E} points in the +x direction.

2) Force on the test charge:

\vec{F} = q\vec{E}

Because q