Electric field, the field of a point charge and field lines
The electric field is the force per unit positive charge, E = F/q₀; for a point charge E = kq/r². Field lines never cross.
What is an electric field?
Faraday gave the idea of an electric field to explain how charges act on each other across space. A charge sets up a field around it. Another charge placed in this field feels a force. The electric field E at a point is the force on a small positive test charge q₀ placed there, divided by q₀: E = F/q₀. Its unit is N C⁻¹. E is a vector, in the direction of the force on a positive charge. The test charge must be very small, so that it does not disturb the charges that make the field. This is like g = F/m for gravity. Once E is known, the force on any charge q there is F = qE. For many charges, add their fields as vectors.
Derivation: field due to a point charge
- 1Place a point charge q at the origin O. Choose a point P at distance r from O.
- 2Put a small positive test charge q₀ at P.
- 3By Coulomb's law, the force on q₀ is F = (1/4πε₀)(q q₀/r²) r̂, where r̂ is the unit vector from O to P.
- 4Field = force per unit test charge: E = F/q₀.
- 5★So, E = (1/4πε₀)(q/r²) r̂. It points radially outward for q > 0 and inward for q < 0. Its size is the same at all points on a sphere centred on q.
★ key step
Properties of electric field lines
- Field lines start on a positive charge and go out radially; for a single charge they end at infinity.
- Field lines come in from infinity and end on a negative charge.
- For a dipole, the lines start on +q and end on −q.
- The tangent to a line at any point gives the direction of E there.
- The number of lines crossing unit area at right angles to them is proportional to E: crowded lines mean a strong field.
- Two field lines never cross. At a crossing, E would have two directions at one point, which is impossible.
- A uniform field is shown by parallel, equally spaced straight lines, as between two oppositely charged plates.
- Field lines are imaginary: they are only a picture of the field.
How to draw field lines for five standard cases
- 1Single positive charge: a dot marked +q with 8 straight lines going radially outward, arrows pointing away.
- 2Single negative charge: a dot marked −q with 8 straight lines coming radially inward, arrows pointing towards it.
- 3Electric dipole (+q and −q close together): lines leave +q, curve round and enter −q. Draw one straight line along the axis between them. The lines are crowded between the charges.
- 4Two equal positive charges: lines leave both charges and bend away from each other. Leave a gap at the midpoint and label it neutral point (E = 0).
- 5Uniform field: parallel, equally spaced straight lines with all arrows the same way, as between two oppositely charged parallel plates.
Worked example: field and force
Find the electric field at a point 20 cm from a charge of +8 nC in air. What force would an electron feel there? (e = 1.6 × 10⁻¹⁹ C)
- Given: q = 8 × 10⁻⁹ C, r = 0.20 m.
- E = kq/r² = 9 × 10⁹ × 8 × 10⁻⁹ ÷ (0.20)² = 72 ÷ 0.04 = 1800 N C⁻¹.
- Direction: away from the positive charge.
- Force on the electron: F = eE = 1.6 × 10⁻¹⁹ × 1800 = 2.88 × 10⁻¹⁶ N.
- The electron is negative, so the force points towards the charge.
E = 1.8 × 10³ N C⁻¹, away from the charge; force on the electron ≈ 2.9 × 10⁻¹⁶ N, towards the charge
Worked example: field at a midpoint
Charges of +3 μC and −3 μC are at A and B, 20 cm apart in air. Find the electric field at M, the midpoint of AB.
- Each charge is 0.10 m from M.
- Field of +3 μC at M: E₁ = 9 × 10⁹ × 3 × 10⁻⁶ ÷ (0.10)² = 2.7 × 10⁶ N C⁻¹, away from A, that is towards B.
- Field of −3 μC at M: E₂ = 2.7 × 10⁶ N C⁻¹, towards B (towards the negative charge).
- Both point the same way, so they add: E = 2 × 2.7 × 10⁶ = 5.4 × 10⁶ N C⁻¹.
5.4 × 10⁶ N C⁻¹, pointing from A towards B
Thermal power stations and brick kilns use electrostatic precipitators: a strong electric field charges the ash in the smoke and pulls it onto the walls, so cleaner air leaves the chimney.
1 marks: “What is the SI unit of electric field?” Newton per coulomb, N C⁻¹.
2 marks: “Why can two electric field lines never cross each other?” The tangent to a field line gives the direction of E. At a crossing there would be two tangents, so E would have two directions at one point, which is impossible.
2 marks: “A proton is placed in a uniform field of 5.0 × 10³ N C⁻¹. Find its acceleration. (e = 1.6 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg)” F = eE = 1.6 × 10⁻¹⁹ × 5.0 × 10³ = 8.0 × 10⁻¹⁶ N. a = F/m = 8.0 × 10⁻¹⁶ ÷ 1.67 × 10⁻²⁷ ≈ 4.8 × 10¹¹ m s⁻², along the field.
E is F per unit plus: the field is the push on a unit positive charge. Lines leave plus, land on minus, and never cross.