Electric Flux, Area Vector & Total Electric Flux

(Class 12 Physics – NCERT )

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1️⃣ Electric Flux – Meaning and Idea

Electric flux gives a measure of how much electric field passes through a given surface.

Think of electric field lines as invisible threads:

• More lines passing through a surface → greater flux

• Fewer lines → smaller flux

👉 Electric flux does not mean electric field is flowing like water.
It is only a mathematical quantity used to describe the electric field.

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2️⃣ Area Vector – Why Area is a Vector

The effect of an electric field on a surface depends not only on:

• Size of the surface
  but also on

• Orientation of the surface

So, area is represented as a vector.

✔ Definition of Area Vector

• Magnitude = area of the surface

• Direction = perpendicular (normal) to the surface

$$\vec{A}=A\widehat{n}$$

where

• $A$= area

• $\widehat{n}$ = unit vector normal to the surface

Important Convention

• For a closed surface, the area vector always points outward

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3️⃣ Electric Flux Through a Plane Surface

If:

• Electric field = $\vec{E}$

• Area vector = $\vec{A}$

• Angle between $\vec{E}$ and $\vec{A}$ = $\theta$

Then electric flux is defined as:

$$\boxed{{\displaystyle \mathrm{\Phi }=\vec{E}\cdot \vec{A}=EA\cos \theta }}$$

This is a dot product, so orientation matters.

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4️⃣ Special Cases of Electric Flux (Very Important)

🔹 Case 1: Maximum Flux

• $\theta =0^{\circ }$

• Surface is perpendicular to electric field

$$\mathrm{\Phi }=EA$$

➡ Maximum number of field lines pass through the surface.

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🔹 Case 2: Zero Flux

• $\theta ={90}^{\circ }$

• Surface is parallel to electric field

$$\mathrm{\Phi }=0$$

➡ Field lines do not cross the surface.

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🔹 Case 3: Inclined Surface

• $0^{\circ }<\theta <{90}^{\circ }$

$$\mathrm{\Phi }=EA\cos \theta$$

➡ Only the effective area $A\cos \theta$ contributes.

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5️⃣ Electric Flux Through a Curved Surface

For a curved or irregular surface, we divide it into many small area elements $dA$.

Flux through a small element:

$$d\mathrm{\Phi }=\vec{E}\cdot d\vec{A}$$

Total electric flux:

$$\boxed{{\displaystyle \mathrm{\Phi }=\int \vec{E}\cdot d\vec{A}}}$$

This is the most general definition of electric flux.

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6️⃣ Total Electric Flux Through a Closed Surface

A closed surface completely encloses a volume (sphere, cube, cylinder).

Sign Convention

• Field lines leaving the surface → positive flux

• Field lines entering the surface → negative flux

👉 Total electric flux is the algebraic sum of flux over the entire closed surface.

This concept leads directly to Gauss’s Law.

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7️⃣ SI Unit of Electric Flux

$$\text{Electric flux}={\text{N m}}^2{\text{ C}}^{-1}$$

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8️⃣ Key Exam Points (One-Look Revision)

• Electric flux is a scalar quantity

• Depends on:

  • Electric field

  • Area

  • Orientation of surface

• Independent of:

  • Shape of surface (for uniform field)

• Area vector is always normal to the surface

• For closed surfaces, area vector is outward

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9️⃣ One-Line Definitions

Electric Flux:

Electric flux through a surface is the surface integral of the normal component of electric field over that surface.

Area Vector:

Area vector of a surface has magnitude equal to the area and direction normal to the surface.