Field lines and electric flux
Electric field lines
Drawing a field arrow at every point gets messy fast. Faraday had a better picture: electric field lines. A field line is a curve whose tangent at every point gives the direction of E there.
For a positive point charge, the field lines are straight lines going radially outwards. For a negative charge, they point radially inwards.
The lines also show the strength of the field. Near the charge they are crowded together, where the field is strong. Further out they spread apart, where it is weak.
Draw a sphere of radius r round the charge. The same number of lines crosses it, but they share an area of 4πr². So the number of lines per unit area falls as 1 over r², just like the field itself.
The rule: the number of lines crossing a unit area held at right angles to the field is proportional to the field strength there.
Field lines start on positive charges and end on negative charges. If there is only one charge, they start or end at infinity. The number of lines drawn is proportional to the charge.
In a region with no charge, field lines are smooth, continuous curves, without breaks.
Two field lines can never cross. If they did, the field at that point would have two directions at once, which is impossible.
And electrostatic field lines never form closed loops. That follows from the conservative nature of the electric field.
Look at two pictures. For two equal positive charges, the lines bend away from each other, and there is a point midway where the field is zero. For a positive and a negative charge, the lines run from one to the other.
Electric flux
Think of water flowing with speed v through a small flat loop of area dS. Face-on to the flow, the volume passing per second is v dS. Tilt the loop, and less water gets through: v dS cos θ.
We define electric flux the same way. Give the small area a direction: a vector ΔS whose size is the area and which points along the normal n̂ to the surface.
The electric flux through it is ΔΦ equals E dot ΔS, which is E ΔS cos θ, where θ is the angle between E and the normal. Flux is a scalar.
In the field-line picture, flux is proportional to the number of lines crossing the surface. Watch the surface turn. Face-on, θ is 0 and the flux is largest: E ΔS.
Edge-on, θ is 90°, no lines cross, and the flux is zero. Turn it further, and the lines enter from the other side: cos θ is negative, so the flux is negative.
For a closed surface, we always take the normal pointing outwards. So flux going out counts as positive, flux coming in as negative.
For a large or curved surface, cut it into small pieces, find E dot ΔS for each, and add them all up. The unit of flux is newton metre squared per coulomb.
Example: a uniform field of 3 × 10³ newtons per coulomb along x. A square of side 10 centimetres has area 0.01 square metre. With its plane parallel to the y-z plane, its normal is along x, so θ is zero.
The flux is 3 × 10³ times 0.01, which is 30 newton metre squared per coulomb. Turn the square so its normal is at 60° to the x-axis: the flux is 30 times cos 60°, which is 15.
Try it: field lines
- The +2 μC charge has twice as many lines as the −1 μC charge. Half of its lines end on the negative charge; the rest go off to infinity.
- Make both charges +1 μC. The lines bend away from each other, and there is an empty spot midway, where E = 0.
- Make them +1 μC and −1 μC: the dipole pattern, with every line running from + to −.
- Drag the charges around. However tangled the pattern, the lines never cross.
Try it: flux through a surface
- This is the worked example: E = 3 × 10³ N/C and a 10 cm square. Face-on, Φ = 30 N m²/C.
- Drag the red knob (or the θ slider) to 60°: the flux halves to 15 N m²/C, and fewer lines cross.
- At 90° the surface is edge-on and no line crosses it. Past 90°, the flux turns negative.
- Double E: the lines get twice as dense, and twice as many cross the surface.
Field line rules
Which of these statements about electrostatic field lines is FALSE?
- They start on positive charges and end on negative charges
- They are crowded together where the field is strong
- Two field lines can cross where the fields of two charges meet
- They do not form closed loops
Answer
C. Field lines never cross. At a crossing, the tangents would give two directions for E at one point, which is impossible. Where two charges’ fields meet, they add into one resultant field, and the lines bend to follow it.
Tilted square
A uniform field of 2 × 10³ N/C points along x. What is the flux through a square of side 20 cm whose normal makes 60° with the x-axis?
- 40 N m²/C
- 80 N m²/C
- 69 N m²/C
- 200 N m²/C
Answer
A. ΔS = (0.2 m)² = 0.04 m². Φ = E ΔS cos θ = 2 × 10³ × 0.04 × cos 60° = 80 × 0.5 = 40 N m²/C. (80 forgets the cos θ; 69 uses sin 60°; 200 forgets to square the side.)
Summary
- Flux through a small area:
ΔΦ = E · ΔS = E ΔS cos θ - Total flux:
Φ = Σ E · ΔS - Unit of flux:
N m² C⁻¹
- A field line is a curve whose tangent gives the direction of E; the density of lines shows its strength.
- Field lines start on + charges and end on − charges, are continuous in charge-free space, never cross, and do not form closed loops.
- Electric flux measures the number of field lines crossing a surface. It is largest face-on (θ = 0°) and zero edge-on (θ = 90°).
- For a closed surface the normal points outwards: outgoing flux is positive, incoming flux negative.