In this lecture, we apply Gauss’s Law to calculate electric fields in important symmetrical charge distributions. These applications are frequently tested in board and entry-test exams.

⚡ Electric Field Inside a Conductor

Using Gauss’s Law, we analyze what happens inside a conductor in electrostatic equilibrium.

🔹 Key Results:

• Electric field inside a conductor is zero

  $$E=0$$

• Excess charge resides only on the surface of the conductor

• Electric field just outside the conductor is perpendicular to the surface

📌 Explanation Using Gauss’s Law:

If we draw a Gaussian surface inside a conductor:

∮E⃗⋅dA⃗=Qencε0\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}∮E⋅dA=ε0​Qenc​​

Since no net charge exists inside the conductor,

Qenc=0⇒E=0Q_{enc} = 0 \Rightarrow E = 0Qenc​=0⇒E=0

This proves that the electric field inside a conductor is zero.

📄 Electric Field Due to an Infinite Sheet of Charge

Now we consider a uniformly charged infinite plane sheet with surface charge density $\sigma$.

Using a cylindrical Gaussian surface (pillbox):

ΦE=2EA\Phi_E = 2EAΦE​=2EA

Applying Gauss’s Law:

2EA=σAε02EA = \frac{\sigma A}{\varepsilon_0}2EA=ε0​σA​ E=σ2ε0E = \frac{\sigma}{2\varepsilon_0}E=2ε0​σ​

📌 Important Result:

• Electric field due to an infinite sheet is constant

• It does not depend on distance from the sheet

⚡ Electric Field Between Two Oppositely Charged Plates

When two large parallel plates carry equal and opposite charges:

• Field due to one plate:

  E=σ2ε0E = \frac{\sigma}{2\varepsilon_0}E=2ε0​σ​

• Inside between plates, fields add up:

E=σε0E = \frac{\sigma}{\varepsilon_0}E=ε0​σ​

• Outside the plates, fields cancel out:

$$E=0$$

📌 Key Observations:

• Electric field between plates is uniform

• This principle is used in capacitors

🎯 By the End of This Lecture

You will:

• Apply Gauss’s Law to symmetric charge distributions

• Prove that electric field inside a conductor is zero

• Derive the field due to an infinite sheet of charge

• Calculate electric field between parallel plates

This lecture strengthens your conceptual understanding and prepares you for capacitor-related topics in the next lessons.