Representation of the Electric Field by Lines: MDCAT Physics notes

What field lines show

Faraday introduced electric lines of force as imaginary lines that picture an electric field. The tangent to a line at any point gives the direction of the electric field $\vec{E}$ there, which is the direction of the force on a small positive test charge.

Properties of field lines

• They start on positive charges and end on negative charges (or at infinity).
• The tangent at any point gives the direction of $\vec{E}$.
• Two field lines never cross; otherwise the field would have two directions at one point.
• Where lines are closer together, the field is stronger; the number of lines per unit area perpendicular to them is proportional to $E$.
• The number of lines leaving or entering a charge is proportional to the size of the charge.
• Lines meet the surface of a conductor at right angles, and there are no lines inside a charged conductor.

Standard patterns

Charge arrangement	Pattern of lines
Isolated positive charge	Straight lines radially outward in all directions
Isolated negative charge	Straight lines radially inward
Equal and opposite charges (dipole)	Curved lines from + to −; field is strong between them
Two equal like charges	Lines repel each other; a neutral point ($E = 0$) midway
Oppositely charged parallel plates	Parallel, equally spaced lines: uniform field

Force, acceleration and path

A charge $q$ in a field feels $\vec{F} = q\vec{E}$, so its acceleration $\vec{a} = q\vec{E}/m$ always points along the tangent to the field line at its position:

• for a positive charge, in the direction of $\vec{E}$;
• for a negative charge, opposite to $\vec{E}$, but still along the tangent.

In a uniform field (straight lines) a charge released from rest moves along a straight line parallel to the field. In a non-uniform field with curved lines, however, the particle does not follow the line of force. Once it has velocity, its inertia carries it off the curve, while the force keeps pointing along the local tangent. So the correct statement is that its acceleration is tangential to the line of force at every instant, not that it moves along the line.

Key formulas

• $\vec{E} = \vec{F}/q_0$
• Point charge: $E = \dfrac{kq}{r^2}$, $k = 9 \times 10^9\ \mathrm{N\,m^2\,C^{-2}}$
• $\vec{a} = q\vec{E}/m$

Common MDCAT traps

• A charge in a curved field does not trace the field line; only its acceleration is along the tangent.
• For a positive charge, lines go radially away, not towards it and not horizontal or vertical.
• Field lines are imaginary and never intersect.
• Crowded lines mean a strong field; widely spaced lines mean a weak one.

Quick revision

• Lines run from + to −.
• Tangent to a line gives the direction of $\vec{E}$.
• Line density is proportional to field strength.
• Parallel equally spaced lines: uniform field.
• Acceleration of a charge is along the tangent to the line.