represent an electric field by means of field lines

Electric Fields and Field‑Line Representation

Learning Objective

Students will be able to:

• State the definition of electric field and the working form F = q E.
• Draw and interpret electric field‑line diagrams for a range of charge configurations.
• Relate line density to field magnitude and use the quantitative relationships required by Cambridge 9702 §18.
• Apply the superposition principle and the uniform‑field formula E = ΔV/d to solve exam‑style questions.

1. Fundamental Definitions & Equations

Concept	Formula	Notes (syllabus emphasis)
Electric field (definition)	\[\mathbf{E}=\frac{\mathbf{F}}{q_0}\]	Force \(\mathbf{F}\) on a *positive* test charge \(q_0\) that is sufficiently small not to disturb the source charges.
Working form used in calculations	\[\mathbf{F}=q\,\mathbf{E}\]	Emphasise the sign convention: a positive test charge moves in the direction of \(\mathbf{E}\); a negative charge experiences a force opposite to \(\mathbf{E}\).
Coulomb’s law (point charges)	\[\mathbf{F}_{12}=k\,\frac{q_1q_2}{r_{12}^{2}}\;\hat{\mathbf r}_{12}, \qquad k=8.99\times10^{9}\ \mathrm{N\,m^{2}\,C^{-2}}\]	Resulting field of a single charge \(Q\): \(\displaystyle \mathbf{E}=k\frac{Q}{r^{2}}\;\hat{\mathbf r}\).
Superposition principle	\[\mathbf{E}_{\text{net}}=\sum_i \mathbf{E}_i\]	Vector sum of the fields produced by each individual charge.
Uniform electric field (parallel‑plate capacitor)	\[E=\frac{\Delta V}{d}\]	\(\Delta V\) = potential difference between the plates, \(d\) = separation. Valid only in the central region where edge effects are negligible (plate dimensions ≫ \(d\)).
Field‑line density ↔ field strength	\[|\mathbf{E}|\propto\frac{N}{A}\]	Number of lines \(N\) crossing a small area \(A\) is proportional to the flux. A common convention is 1 × 10⁶ lines per coulomb. The rule is used qualitatively in the syllabus.

2. Rules for Drawing Electric Field Lines

Rule	What the syllabus expects
Start and end on charges (or at infinity)	Lines originate on positive charges and terminate on negative charges. For an isolated charge they begin or end at infinity. Lines never start or end inside a conductor.
Direction of the field	The tangent to a line gives the direction of \(\mathbf{E}\): away from +, toward –.
Density ↔ magnitude	Closer spacing ⇒ stronger field. The number of lines drawn from a charge is proportional to \(|Q|\). Example: a charge \(2Q\) is represented by twice as many lines as a charge \(Q\).
No crossing	Field lines never intersect; an intersection would give two possible directions for \(\mathbf{E}\) at that point.
Symmetry	Use spherical, cylindrical or planar symmetry to decide the overall pattern before drawing individual lines.
Superposition	For several charges, first sketch the individual patterns, then combine them by vector addition of \(\mathbf{E}\). The final diagram must reflect the resultant direction and relative strength.

3. Typical Configurations

3.1 Single Point Charge

• Positive charge \(+Q\): radial lines radiate outward uniformly; line density ∝ \(Q\).
• Negative charge \(-Q\): lines converge inward uniformly.
• Isolated charge → lines start (or end) at infinity.

3.2 Electric Dipole

• Two equal and opposite charges \(\pm Q\) separated by distance \(d\).
• Lines emerge from \(+Q\), curve, and terminate on \(-Q\); the pattern is symmetric about the axis joining the charges.
• Near the centre the field is approximately uniform and directed from + to –; far away (\(r\gg d\)) the field falls off as \(1/r^{3}\) (dipole field).

3.3 Uniform Field (Parallel‑Plate Capacitor)

• Large, parallel, oppositely charged plates.
• Field lines are straight, parallel, equally spaced, directed from the positive plate to the negative plate.
• Magnitude given by \(E=\Delta V/d\); only the central region (away from edges) satisfies this relation.

3.4 Conductors in Electrostatic Equilibrium

• Inside a conductor \(\mathbf{E}=0\) → no field lines within the material.
• Field lines meet the surface **perpendicularly** (normal to the surface).
• All excess charge resides on the outer surface; the surface charge density is proportional to the local line density.
• Lines never start or end inside the conductor; they begin/terminate on the surface charge.

4. Using Field‑Line Diagrams Quantitatively

In Cambridge exams the line‑density rule is mainly used to justify qualitative statements such as “the field is stronger where the lines are closer”. When a question supplies a specific convention (e.g., \(1\times10^{6}\) lines / C), the field magnitude at a point can be estimated by:

\[ |\mathbf{E}|\approx\frac{N_{\text{cross}}}{A}\times\left(\frac{1\ \text{C}}{10^{6}\ \text{lines}}\right) \]

where \(N_{\text{cross}}\) is the number of lines crossing a small imaginary surface of area \(A\) centred on the point of interest.

5. Worked Examples (Cambridge‑style)

Example 1 – Direction of the Field at the Mid‑point of a Dipole

Question: 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 state the direction (not the magnitude) of the electric field at the midpoint.

1. Identify the configuration – an electric dipole.
2. Draw radial lines from the positive charge and terminating on the negative charge; the pattern is symmetric about the line joining them.
3. At the midpoint the radial components along the joining line cancel (equal magnitude, opposite direction).
4. The remaining components are perpendicular to the joining line, pointing from the positive charge toward the negative charge in the plane that contains the charges. Result: the net field is horizontal (if the charges are aligned vertically) and directed from \(+Q\) toward \(-Q\).

Example 2 – Magnitude of the Field of a Single Point Charge

Question: A point charge \(Q=+5\;\text{nC}\) is isolated in vacuum. Calculate the electric field magnitude at a distance \(r=4.0\;\text{cm}\) from the charge.

\[ E = k\frac{Q}{r^{2}} = 8.99\times10^{9}\,\frac{5\times10^{-9}}{(0.04)^{2}} \approx 2.8\times10^{4}\ \text{N C}^{-1} \]

The field‑line diagram would show radial lines outward; the line density at \(r=4\ \text{cm}\) is proportional to this value.

Example 3 – Uniform Field Between Parallel Plates (Linking ΔV and E)

Question: Two large parallel plates are separated by \(d=2.0\;\text{mm}\) and a potential difference of \(V=150\;\text{V}\) is applied. Determine the magnitude of the electric field between the plates and describe the corresponding field‑line picture.

\[ E = \frac{\Delta V}{d}= \frac{150\ \text{V}}{2.0\times10^{-3}\ \text{m}} = 7.5\times10^{4}\ \text{N C}^{-1} \]

Field lines are straight, parallel, equally spaced, and directed from the positively charged plate to the negatively charged plate. The uniform‑field formula is valid only in the central region where edge effects are ignored (plate dimensions ≫ \(d\)).

Example 4 – Field‑Line Density for Different Charge Magnitudes

Question: On a diagram, a charge \(+Q\) is represented by 8 field lines. How many lines should be drawn for a charge \(+2Q\) and for a charge \(-\tfrac{1}{2}Q\)?

• \(+2Q\): double the magnitude → \(2\times8 = 16\) lines, all pointing away from the charge.
• \(-\tfrac{1}{2}Q\): half the magnitude and opposite sign → \(\tfrac{1}{2}\times8 = 4\) lines, all pointing toward the charge.

6. Summary Checklist (Exam Quick‑Reference)

• Definition: \(\displaystyle \mathbf{E}=\frac{\mathbf{F}}{q_0}\) and \(\displaystyle \mathbf{F}=q\,\mathbf{E}\) (positive test charge).
• Field lines start on + charges, end on – charges, or at infinity; they never start/end inside a conductor.
• Direction of \(\mathbf{E}\) = tangent to a line; density ↔ magnitude (more lines → stronger field).
• Lines never cross.
• Use symmetry (spherical, cylindrical, planar) to decide the overall pattern.
• Superposition: draw individual patterns, then combine by vector addition.
• Uniform field between parallel plates: \(E=\Delta V/d\); straight, parallel, equally spaced lines.
• Inside a conductor \(\mathbf{E}=0\); lines meet the surface perpendicularly.
• Quantitative line‑density rule is qualitative in the syllabus; actual calculations use Coulomb’s law or \(E=\Delta V/d\).