Step-by-Step Derivation of Electric Field Using Gauss’s Law
Electric Field Due To Infinite Plane is a classic electrostatics topic, vital for JEE Main. It explains the nature of the field created by an infinite sheet of charge and helps solve related numerical problems easily. Many real-life and exam problems approximate large flat surfaces as infinite planes to simplify calculations and reveal the underlying physics. You’ll often use this concept to analyze fields around plates of capacitors and charged surfaces.
One key feature of the electric field due to an infinite plane is its uniformity: the magnitude is the same anywhere near the surface and points perpendicular to the plane. This powerful idea rests on the symmetry of the setup and allows for fast calculation using Gauss’s law. Both the formula and its derivation are standard, so JEE students must know them very well.
In JEE Main and similar exams, the infinite plane model makes it possible to test your understanding of electric fields, superposition, and uniform charge distributions. The formula and methods discussed here also form the basis for tackling more complex setups like two parallel infinite plates.
Physical Meaning and Visualization of Electric Field Due To Infinite Plane
Picture an infinitely wide flat sheet: the electric field due to infinite plane sheet consists of straight, equally spaced lines on both sides of the sheet. The lines emerge perpendicularly away from the surface if the sheet is positively charged, or towards the surface if the charge is negative.
Unlike a point charge or finite plate, the field does not weaken with distance. This is because, at any point near the sheet, the contributions from parts of the sheet far away are balanced symmetrically, ensuring uniform field strength. This makes infinite planes an excellent approximation for very large surfaces compared to your distance from them.
This uniformity is a distinctive feature when comparing the electric field due to infinite plane with fields due to other charge geometries such as lines or spheres. To strengthen your grasp, review the direction and pattern of electric field lines in the electric field lines topic.
Derivation of Electric Field Due To Infinite Plane Using Gauss’s Law
- Start with a non-conducting infinite sheet carrying uniform surface charge density σ (C/m²).
- Select a Gaussian surface: a cylinder (“pillbox”) with flat faces parallel to the sheet, passing through it.
- Due to symmetry, Gauss’s law applies. The field is perpendicular to the sheet and has equal magnitude on both sides.
- The total electric flux through the cylinder is 2EA (since the field passes through both ends: each of area A).
- Total enclosed charge is σA.
- By Gauss’s law: \(2EA = \frac{\sigma A}{\varepsilon_0}\)
- Solve for E:
\(E = \frac{\sigma}{2\varepsilon_0}\)
This stepwise derivation is standard for JEE Main, so remember the symmetry argument and choice of surface.
Electric Field Due To Infinite Plane: Formula, Units, and Terms
| Formula | Symbol | Physical Meaning | SI Unit |
|---|---|---|---|
| \(E = \frac{\sigma}{2\varepsilon_0}\) | E | Electric field due to infinite sheet | N/C, or V/m |
| — | σ | Surface charge density | C/m² |
| — | \(\varepsilon_0\) | Permittivity of free space | \(8.85 \times 10^{-12}\) C²/(N·m²) |
Field direction: Perpendicular to the surface, away from positive charges, toward negative charges. The value does not depend on how far you are from the plane.
The formula also appears in problems involving charged shells and, with sign adjustments, opposing plates.
Special Cases: Conducting, Non-Conducting, and Two Parallel Sheets
The electric field due to infinite plane sheet formula adapts to common JEE cases.
| Case | Formula | Features/Notes |
|---|---|---|
| Non-conducting infinite sheet | \(E = \frac{\sigma}{2\varepsilon_0}\) | Field equal on both sides |
| Conducting infinite sheet | \(E = \frac{\sigma}{\varepsilon_0}\) | All charge on one side, field exists only outward |
| Between two parallel sheets (charges +σ, –σ) | \(E = \frac{\sigma}{\varepsilon_0}\) | Fields add between, cancel outside |
For a conducting sheet, charges collect at the surface, doubling the field on the side with charge, and leaving zero field inside. For two parallel plates with equal-magnitude, opposite charges, the resulting field between the plates becomes uniform, a crucial idea in capacitor problems.
Carefully apply sign conventions and confirm sheet type in JEE problem statements.
Numerical Example and Applications in JEE Main
Let’s solve a classic numerical using the electric field due to infinite plane formula.
Example: A large non-conducting surface has uniform surface charge density σ = 3 × 10–6 C/m². Find the electric field at a point near the surface. (ε0 = 8.85 × 10–12 C²/N·m²)
Using \(E = \frac{\sigma}{2\varepsilon_0}\):
\(E = \frac{3 \times 10^{-6}}{2 \times 8.85 \times 10^{-12}} = \frac{3 \times 10^{-6}}{1.77 \times 10^{-11}}\)
\(E = 1.695 \times 10^{5}\) N/C
Final field: 1.70 × 105 N/C (toward or away based on sign of σ)
Beyond JEE numericals, this formula underpins calculations in parallel plate capacitors and fields near charged surfaces of large conductors. Review its relevance in electric flux area vector and electrostatics mock tests.
- The field is always perpendicular to the sheet.
- It does not depend on distance from the sheet.
- Careful with signs: direction reverses for negative charge.
- Model “very large” sheets as infinite when your distance is small versus the sheet's dimensions.
- Always check if asked about field outside, inside, or between plates.
Summary, Revision, and Key Formulae for Electric Field Due To Infinite Plane
- The field due to an infinite plane sheet is uniform and points perpendicularly.
- Main formula: \(E = \dfrac{\sigma}{2\varepsilon_0}\) for non-conducting, \(E = \dfrac{\sigma}{\varepsilon_0}\) for conducting sheets.
- Use Gauss’s law and symmetry for fast derivations in exams.
- Does not decrease with distance. Critical in electric charges and fields concepts.
- Appears in mock tests, revision, and important questions in Vedantu resources.
| Sheet Type | Formula | Direction |
|---|---|---|
| Infinite (non-conducting) | \(E = \frac{\sigma}{2\varepsilon_0}\) | Perpendicular, both sides equal |
| Infinite conducting | \(E = \frac{\sigma}{\varepsilon_0}\) | Perpendicular, outward only |
| Two plates (+σ, –σ) | \(E = \frac{\sigma}{\varepsilon_0}\) (between sheets) | From + to – plate |
Explore related topics on Vedantu for further mastery, such as electric field intensity and Coulomb’s law. Practicing these derivations and problems boosts your speed and confidence for JEE Main success.
For ongoing exam preparation with the latest patterns and solved examples, use other Vedantu Physics resources. Deepening your command of electric field due to infinite plane gives you a real edge in electrostatics.
FAQs on Electric Field Due to an Infinite Plane Sheet of Charge
1. What is the formula for the electric field due to an infinite plane sheet of charge?
The formula for the electric field due to an infinite plane sheet of charge is:
E = σ / (2ε₀), where:
- σ = surface charge density in C/m²
- ε₀ = permittivity of free space (8.854 × 10⁻¹² C²/N·m²)
2. How is the electric field due to an infinite plane sheet derived using Gauss’s Law?
To derive the electric field using Gauss’s Law, follow these steps:
- Consider an infinite plane with surface charge density σ.
- Choose a Gaussian surface as a cylinder (pillbox) perpendicular to the plane.
- The flux passes through both flat faces, so total flux = 2EA.
- According to Gauss’s Law: 2EA = σA / ε₀.
- Solve for E: E = σ / (2ε₀).
3. Does the electric field from an infinite plane sheet depend on distance?
No, the electric field due to an infinite plane sheet of charge is independent of distance.
This result comes directly from Gauss’s Law and the symmetry of the infinite plane. The key points:
- The field's magnitude stays constant on either side of the sheet, no matter how far you move from it.
- This is because a truly infinite plane has no edges; every location 'sees' the same amount of charge in all directions.
- In exams, always use the direct formula, regardless of distance from the sheet.
4. What is the difference between the electric field of a conducting and non-conducting infinite sheet?
The main difference is in the distribution of charge and field direction:
- Conducting Sheet: Charge resides only on the surface; the field outside is E = σ / ε₀ (for one side only), and zero inside the conductor.
- Non-conducting (insulating) Sheet: Charge can be distributed throughout; field both sides is E = σ / (2ε₀).
- This distinction sometimes comes in numerical questions and conceptual MCQs.
5. What is the electric field between two infinite parallel sheets of charge?
The net electric field between two parallel infinite sheets depends on the sign and value of their surface charge densities.
For sheets with charge densities σ₁ and σ₂:
- If parallel and carrying equal but opposite charges, the field between them becomes E = σ / ε₀ and outside it is zero.
- The direction of the field is from the positively charged sheet to the negatively charged sheet.
- General case: The net field at any region is the vector sum of the fields due to each sheet.
6. Why does the electric field due to an infinite plane sheet remain constant at any distance?
The electric field is constant because an infinite sheet has perfect symmetry and produces parallel, uniformly spaced field lines.
This means:
- Every point at equal distance from the plane experiences the same amount of field.
- The 'infinite' size ensures the ratio of area to distance doesn't change.
- This concept holds for practical purposes when the plane is very large compared to the observation distance.
7. How can you visually represent the electric field near an infinite plane sheet of charge?
The electric field near an infinite plane sheet is shown by equally spaced, parallel lines perpendicular to the sheet.
- Field lines point away from the sheet for positive charge and towards for negative charge.
- Spacing between lines indicates field uniformity (same everywhere).
- There are no curved lines, as from point sources.
8. What are the applications of the electric field due to an infinite plane sheet?
Some key applications include:
- Calculating electric fields in parallel plate capacitors (used in electronics).
- Modeling charge distribution and field near large charged surfaces.
- Simplifying analysis in problems where surface size >> distance from observer (JEE/Boards questions).
- Understanding shielding and field strength in conducting sheets.
9. How do you solve numerical problems involving the electric field of an infinite plane sheet?
To solve numericals on this topic:
- Write the formula: E = σ / (2ε₀) (for non-conducting sheet) or E = σ / ε₀ (for outside a conducting sheet).
- Identify σ (surface charge density), convert units if needed.
- Plug values directly; no need to use distance variable.
- For multiple sheets, add/subtract fields as vectors based on direction.
10. What is the significance of surface charge density (σ) in the electric field of a plane sheet?
Surface charge density (σ) determines the amount of electric field produced by the sheet.
- It is defined as the charge per unit area (C/m²).
- The larger the σ, the stronger the electric field according to E = σ / (2ε₀).
- σ is a standard parameter in all exam formulas and numericals involving plane sheets.
