Electric Flux

A measure of electric field through a surface, given by the number of electric lines of force per unit area enclosing the electric lines of force, is called electric flux.

OR

Electric flux through a surface is defined as the dot product of the electric field vector and the area vector of the surface.

For any general surface,

E = \[\frac {\text {Number of electric lines of force}}{\text {Area enclosing the electric lines of force}}\]

OR

Φ = EA cos θ

where:

• Φ = electric flux
• E = magnitude of the electric field
• A = area of the surface
• θ = angle between \[\vec{E}\] and the area vector \[\vec{E}\]

SI Unit

• SI unit of electric flux = N m² C⁻¹
• Equivalent SI unit = V m

Dimensional Formula: [ML³T^-3A^-1]

Electric flux is the measure of the electric field passing through a given surface. In the field-line picture, it is proportional to the number of electric field lines crossing the surface.

Imagine rain falling vertically on a sheet of cardboard. If the cardboard is held flat, maximum rain passes through its effective area; if it is tilted, less rain passes through; if it is turned edge-on, almost no rain passes through.

This is exactly how electric flux works: the orientation of the surface matters, not just its area. Flux tells how effectively a surface “captures” the electric field.

To calculate flux, a surface is assigned an area vector. The magnitude of this vector equals the area of the surface, and its direction is normal to the surface.

Important Convention

• For an open surface, either normal may be chosen, but the choice must remain consistent.
• For a closed surface, the area vector is always taken to point outward along the outward normal.
• Electric flux depends on the angle between the electric field and the area vector, not between the field and the surface itself.

The value of flux changes with orientation.

Case 1: Surface Normal to Electric Field

If the area vector is parallel to the electric field, then θ = 0°, so

This gives maximum positive flux.

Case 2: Surface Parallel to Electric Field

If the electric field is parallel to the surface, then the angle between \[\vec{E}\] and \[\vec{A}\] is 90°. Therefore,

No electric field lines cross the surface.

Case 3: Field Opposite to Area Vector

If \[\theta = 180°\], then

This gives maximum negative flux.

Electric flux through a closed surface leads directly to Gauss's law, one of the most important results of electrostatics.

Electric flux is a scalar quantity because it is the dot product of two vectors. However, its value may be positive, negative, or zero, depending on the field's orientation relative to the chosen area vector.

Sign Rules

• Positive flux: field lines go outward or along the chosen normal.
• Negative flux: field lines enter inward or opposite to the chosen normal.
• Zero flux: field lines are parallel to the surface.

Electric flux depends on:

• the strength of the electric field E,
• the area of the surface A,
• The angle θ between the field and the area vector.

It does not depend only on the area. A large area may still have zero flux if the field is parallel to the surface.