Electric Flux 😊

1. Quick recap of field lines (why flux matters)

  • Field lines begin on positive charges and end on negative charges or at infinity :contentReference[oaicite:0]{index=0}.
  • They form smooth, unbroken curves in regions without charge.
  • No two field lines ever cross; crossing would make the field’s direction ambiguous.
  • Because the electrostatic field is conservative, field lines never make closed loops.

2. From flowing water to electric flux 🚰

Picture a stream of water with speed v hitting a tiny flat patch of area dS that faces the flow. The water volume rushing through each second is v dS. If you tilt the patch by an angle q, only the component dS cos q “catches” the flow, so the volume rate becomes v dS cos q :contentReference[oaicite:1]{index=1}.

Replace water speed v with electric field E and you have the idea of electric flux.

3. Treat area like a vector 📏

  • Give every small patch an area-vector dS =  dS, pointing along the patch’s normal.
  • For curved surfaces, chop them into many tiny flat bits and assign a vector to each bit.
  • For a closed surface, use the outward normal as the direction of every dS :contentReference[oaicite:2]{index=2}.

4. Definition of electric flux 🔄

A small element of flux is

$$\Delta\phi \;=\; \mathbf{E}\cdot d\mathbf{S} \;=\; E\,dS\cos q \tag{1.11}$$

Here q is the angle between the field E and the area-vector dS. If q = 90° the field skims along the surface and no lines pierce it 🙂.

Unit: newton coulomb−1 metre2 (N C−1 m2) :contentReference[oaicite:3]{index=3}.

5. Total flux through any surface

Split the surface into tiny pieces, sum their fluxes, and shrink the pieces to zero size. That limiting sum becomes an integral:

$$ \phi \;=\; \sum \mathbf{E}\cdot\Delta\mathbf{S} \;\longrightarrow\; \oint_S \mathbf{E}\cdot d\mathbf{S} \tag{1.12} $$

We use ≈ when the pieces are still finite; the integral is exact when the pieces become infinitesimally small :contentReference[oaicite:4]{index=4}.

6. Why orientation matters 🧭

The very same ring lets the most “stuff” through when you hold it face-on to the flow. Tip it, and the effective opening shrinks to dS cos q. That is why the flux formula includes the cosine factor.

High-Yield Ideas for NEET 📚

  1. Understand and apply Δφ = E dS cos q for any orientation of the surface patch.
  2. Recognise that the area-vector dS for a closed surface always points outward.
  3. Convert a summation of flux over tiny patches into the surface integral ∮ E·dS quickly.
  4. Relate flux to the density of field lines: more lines per area ⇒ stronger field.