Chapter 1 · Sections 1.9–1.10Textbook pages 23–27

Electric Field Lines and Electric Flux

Arrows at every point are messy. Faraday found a cleaner picture: join the arrows into smooth field lines. On this page you see why the crowding of lines shows the strength of the field (a solid-angle argument), plot the field lines of up to three charges yourself, and then count how many lines pass through a tilted surface – the electric flux.

📖 What you need (pages 1, 2 and 9 of this guide)

Field of a point charge E = (1/4πε₀) Q/r², radially outward for +Q, inward for −Q; fields of several charges add as vectors (page 9).
Plane angle in radians: Δθ = Δl/r (arc ÷ radius). Area of a sphere 4πr² (page 2).
Dot product: A·B = AB cos θ (page 1).

1. From field arrows to field lines (Fig 1.15)

Draw the field of a point charge as arrows: each arrow shows the force on a unit positive charge placed at its tail, and its length shows the strength. The arrows get shorter (∝ 1/r²) as we move out, always pointing radially outward. Now connect the arrows that point in one direction – each chain becomes an electric field line (വൈദ്യുത മണ്ഡലരേഖകൾ).

🎬 Fig 1.15 – field of a point charge: arrows → lines

Have we lost the information about strength that the arrow lengths carried? No. The strength is now shown by the density of lines: near the charge the field is strong and the lines are close together; far away the field is weak and the lines are well separated. Someone else may draw more lines – only the relative density in different regions matters.

2. Why density ∝ 1/r²: the solid angle (Fig 1.16)

Field lines live in three dimensions, so the density means number of lines per unit area perpendicular to the lines. In two dimensions a small arc Δl at distance r subtends a plane angle Δθ = Δl/r. In three dimensions a small perpendicular area ΔS at distance r subtends a solid angle (ഘനകോൺ)

ΔΩ = ΔSr² (unit: steradian, sr; a whole sphere is 4π sr)

The field lines of a point charge are radial, so the same number n of lines fills a given solid angle ΔΩ at every distance. At P₁ (distance r₁) these n lines cross an area r₁²ΔΩ; at P₂ (distance r₂) they cross r₂²ΔΩ:

density at P₁ = nr₁²ΔΩ , density at P₂ = nr₂²ΔΩ ⇒ density ∝ 1r² ∝ E

🧪 Virtual lab: same lines, bigger area (Fig 1.16)

cone half-angle α ° r₂ m

A cone of half-angle α has solid angle ΔΩ = 2π(1 − cos α). The patch it cuts at distance r has area r²ΔΩ. The number of lines inside the cone never changes, so lines per m² falls as 1/r² – exactly like E.

3. Field lines – definition and rules (Fig 1.17)

⭐ Electric field line

An electric field line is a curve drawn so that the tangent at each point gives the direction of the net electric field at that point. An arrow on the curve shows which of the two tangent directions is meant. Field lines are space curves (3-D), though we draw them in a plane. Faraday called them “lines of force”.

Field lines start from positive charges and end at negative charges. With a single charge they may start or end at infinity.
In a charge-free region field lines are continuous curves without breaks.
Two field lines can never cross. (At a crossing the field would have two directions at one point – absurd.)
Electrostatic field lines do not form closed loops (the electrostatic field is conservative – Chapter 2).
Where lines are crowded the field is strong; where they spread out it is weak.

✏️ Interactive field-line plotter – 1 to 3 charges

q₁ q q₂ q q₃ q lines per unit charge

Drag the charges. Each line is traced step by step along the direction of the total field (superposition). The number of lines leaving a charge is proportional to its charge. Buttons (a)–(d) give the four patterns of Fig 1.17. Two equal + charges: see the empty “neutral point” between them, where E = 0 – the lines bend away, showing repulsion. Dipole: lines stretch from + to −, showing attraction.

Think: (a) Why can't a field line have a sudden break? (b) Why can't two field lines cross? (c) Is a field line the path a charge would follow?

(a) The field exists and changes smoothly at every point in a charge-free region, so a test charge placed anywhere along the line feels a force – the line must continue. (b) At a crossing point there would be two tangents, i.e. two directions of E at one point, which is impossible because the net field at a point has one definite direction. (c) Not in general. The line gives the direction of the force (and hence of the acceleration), not of the velocity. A charge released from rest in a uniform field does move along a straight field line, but in a curved field its inertia makes it leave the line.

4. Electric flux (Fig 1.18, 1.19)

Think of water flowing with velocity v through a small ring of area dS. If the ring faces the flow, the volume crossing per second is v dS. If the ring's normal makes an angle θ with v, only the projection dS cos θ counts – hold the ring edge-on and no water passes. For the electric field we define an analogous quantity, the electric flux (വൈദ്യുത ഫ്ലക്സ്) – though nothing actually flows.

The number of field lines crossing a small area ΔS placed normal to E is proportional to E ΔS. Tilt the area by θ and fewer lines cross: only the projection ΔS cos θ is normal to E. At θ = 90° the lines run parallel to the area and none cross. So orientation matters, and we treat area as a vector: the area vector (ഏരിയ സദിശം) ΔS = ΔS n̂ has magnitude ΔS and points along the normal n̂. For a closed surface we always take the outward normal.

🖼 Fig 1.19 – area vector: ΔS = ΔS n̂ (outward normal on a closed surface)

⭐ Definition – electric flux

Δφ = E·ΔS = E ΔS cos θ (1.11)

θ is the angle between E and ΔS (the outward normal for a closed surface). Flux is a scalar; its unit is N C⁻¹ m² (N m²/C). Read it as E × (ΔS cos θ) – E times the projected area – or as (E cos θ) × ΔS – the normal component E_⊥ times the area. Total flux through a surface: φ ≈ ∑ E·ΔS (exact as ΔS → 0, when the sum becomes an integral).

🧪 Virtual lab: tilt the area – flux slider (Fig 1.18)

angle θ between E and n̂ ° E N/C ΔS m²

Edge-on view: the plate is the thick line, n̂ its normal. Highlighted field lines are the ones that pass through it – their number follows cos θ. For θ > 90° the lines enter against n̂, so the flux is negative.

🌍 Real life – a solar panel

A solar panel collects the most sunlight when it faces the Sun squarely (θ = 0) and almost nothing when sunlight grazes along it (θ ≈ 90°). The same cos θ factor decides how much electric flux passes through an area. (Here only the geometry is the same – sunlight is a flow of energy, flux of E is not a flow of anything.)

🌐 Explore online needs internet · opens in a new tab

🧪 Charges and Fields (PhET simulation) PhET, University of Colorado Boulder
Place charges and see the electric field vectors and field of a dipole.
🧪 Electric Field Hockey (PhET simulation) PhET, University of Colorado Boulder
A game: steer a charged puck using the forces of other charges.
📄 Electric Field Lines The Physics Classroom
Rules for drawing field lines, with diagrams.
📄 Electric field Wikipedia
Definition, field of a point charge and superposition.

5. Check yourself

🧭 Summary

A field line is a curve whose tangent gives the direction of E; the density of lines shows the strength of E.
For a point charge the same number of lines fills a solid angle ΔΩ = ΔS/r² at every distance → density ∝ 1/r² ∝ E.
Rules: start on +, end on − (or at infinity); continuous; never cross; no closed loops in electrostatics.
Area vector ΔS = ΔS n̂ (outward normal for a closed surface).
Electric flux Δφ = E·ΔS = EΔS cos θ; scalar; unit N C⁻¹ m²; maximum at θ = 0, zero at θ = 90°, negative beyond 90°.

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