AcademyElectric Charge and Fields
Academy
Field Line Representations
Level 1 - Physics topic page in Electric Charge and Fields.
Principle
Electric field lines are a diagrammatic representation: tangents show field direction and line density shows relative field strength.
Notation
\(\vec E\)
electric field
\(\mathrm{N\,C^{-1}}\)
\(\hat t\)
tangent direction to a field line
1
\(N_{\mathrm{lines}}\)
number of drawn lines assigned to a charge
1
\(q\)
source charge
\(\mathrm{C}\)
\(A\)
small area pierced by field lines
\(\mathrm{m^{2}}\)
Method
Derivation 1: Direction from a tangent
A field line is drawn so that the local tangent points in the electric field direction.
Tangent rule
\[\hat t\parallel\vec E\]
Positive test charge
\[\vec F=q_0\vec E\quad(q_0>0)\]
Arrow direction
\[\text{field-line arrows follow the force on a positive test charge}\]
Derivation 2: Sources, sinks, and relative strength
Lines begin on positive charge and end on negative charge or at infinity. More lines in the same area means a stronger field there.
Positive source
\[q>0\Rightarrow\text{lines leave}\]
Negative source
\[q<0\Rightarrow\text{lines enter}\]
Relative strength
\[|\vec E|\propto\frac{N_{\mathrm{lines}}}{A}\]
Charge size
\[N_{\mathrm{lines}}\propto |q|\]
Derivation 3: Why lines do not cross
At one point in space, the electric field has one vector value. Two crossing field lines would imply two tangent directions for \(\\vec E\) at the same point.
Single field value
\[\vec E(\vec r)\ \text{is unique}\]
Crossing contradiction
\[\hat t_1\ne\hat t_2\Rightarrow\text{two directions at one point}\]
Rules
These are the rules for reading and drawing electric field lines.
Tangent direction
\[\hat t\parallel\vec E\]
Positive charges
\[q>0\Rightarrow\text{lines point outward}\]
Negative charges
\[q<0\Rightarrow\text{lines point inward}\]
Line density
\[|\vec E|\propto N_{\mathrm{lines}}/A\]
Examples
Question
A diagram shows field lines pointing radially inward toward a point charge. What is the sign of the charge?
Answer
Field lines enter negative charges, so the point charge is negative.
Checks
- Field lines are not physical strings or particle paths.
- The field direction is tangent to the line, not necessarily toward the nearest charge.
- Lines never cross in an electrostatic field diagram.
- More densely packed lines mean larger \(|\\vec E|\), not larger charge speed.