represent an electric field by means of field lines

Published by Patrick Mutisya · 14 days ago

Cambridge A-Level Physics 9702 – Electric Fields and Field Lines

Electric Fields and Field Lines

Learning Objective

Represent an electric field by means of field lines and interpret what the lines convey about the field.

Key Concepts

  • An electric field \$\mathbf{E}\$ at a point is the force \$\mathbf{F}\$ experienced by a test charge \$q_0\$ divided by the magnitude of the test charge:

    \$\mathbf{E} = \frac{\mathbf{F}}{q_0}\$

  • Field lines are a visual tool to depict the direction and relative strength of \$\mathbf{E}\$.

  • Field lines are drawn so that a positive test charge would move along them.

Rules for Drawing Electric Field Lines

RuleExplanation
Start and end on chargesLines originate on positive charges and terminate on negative charges (or at infinity for isolated charges).
DirectionAt any point, the tangent to a field line gives the direction of \$\mathbf{E}\$ (away from +, toward –).
DensityThe number of lines per unit area is proportional to the magnitude of \$\mathbf{E}\$; closer lines mean stronger field.
No crossingField lines never intersect because the direction of \$\mathbf{E}\$ would be ambiguous.
SymmetryUse the symmetry of the charge configuration to simplify the pattern (spherical, cylindrical, planar).
Number of linesFor a charge \$Q\$, the total number of lines drawn is proportional to \$|Q|\$; e.g., 1 × 10⁶ lines per coulomb is a common convention.

Typical Configurations

1. Single Point Charge

For a positive charge \$+Q\$, lines radiate outward uniformly; for a negative charge \$-Q\$, they converge inward.

Suggested diagram: Radial field lines for a single positive charge and a single negative charge.

2. Electric Dipole

A dipole consists of equal and opposite charges \$+Q\$ and \$-Q\$ separated by distance \$d\$. Field lines emerge from \$+Q\$, curve around, and end on \$-Q\$.

Near the centre, the field approximates that of a uniform field if the observation point is far compared with \$d\$.

Suggested diagram: Field lines of an electric dipole, showing curvature from +Q to –Q.

3. Uniform Electric Field

A uniform field can be produced between two large parallel plates with opposite charges. Field lines are straight, parallel, and equally spaced, directed from the positive plate to the negative plate.

Suggested diagram: Parallel straight field lines between oppositely charged plates.

4. Conductors in Electrostatic Equilibrium

  • Inside a conductor, \$\mathbf{E}=0\$; therefore, no field lines exist within.

  • Field lines meet the surface of a conductor perpendicularly.

Suggested diagram: Field lines terminating perpendicularly on a charged conducting sphere.

Using Field Lines to Determine Field Strength

If the number of lines crossing a surface of area \$A\$ is \$N\$, the magnitude of the field can be estimated by

\$|\mathbf{E}| \propto \frac{N}{A}\$

In quantitative problems, the proportionality constant is set by the chosen convention for the number of lines per coulomb.

Worked Example

Problem: Two point charges, \$+2\,\mu\text{C}\$ at the origin and \$-2\,\mu\text{C}\$ at \$(0,0,0.10\ \text{m})\$, are placed in free space. Sketch the field lines and determine the direction of the field at the midpoint.

  1. Identify the charges: equal magnitude, opposite sign → dipole.

  2. Draw lines emerging from \$+2\,\mu\text{C}\$ and terminating on \$-2\,\mu\text{C}\$, symmetric about the line joining them.

  3. At the midpoint, the contributions from each charge have equal magnitude but opposite direction along the line joining the charges, so they cancel. The net field is therefore perpendicular to that line, pointing from the positive charge toward the negative charge in the plane that contains the charges.

Suggested diagram: Field line sketch for the dipole with the midpoint field direction indicated.

Summary

  • Field lines give a qualitative picture of the electric field direction and relative magnitude.

  • Follow the five fundamental rules when drawing them.

  • Use symmetry to simplify complex configurations.

  • Remember that field lines are a representation, not a physical entity.