Derivation and Formula for Electric Field of an Infinite Sheet
The electric field due to an infinite plane sheet of charge is a fundamental concept in electrostatics, frequently discussed in JEE Main and advanced physics studies. This field exhibits unique properties, such as uniform magnitude and direction, which are a direct consequence of the infinite extent and symmetry of the plane sheet.
Definition and Physical Meaning
An infinite plane sheet of charge refers to a surface that extends without boundary in all directions, having a uniform surface charge density $\sigma$. The resulting electric field at any point near the sheet is perpendicular to its surface and is uniform in magnitude and direction.
Derivation of Electric Field Using Gauss’s Law
The derivation uses Gauss's law, which links the net electric flux through a closed surface to the total charge enclosed. Due to symmetry, a cylindrical Gaussian surface, often called a pillbox, is chosen perpendicular to the sheet, with its flat faces parallel to the sheet.
Let the surface charge density be $\sigma$ and the area of each flat face of the cylinder be $A$. The electric field, $E$, is directed normally away from the sheet on both sides if the charge is positive, or towards the sheet if negative.
The total electric flux through the Gaussian surface is given by:
$\Phi_E = \text{Flux through top face} + \text{Flux through bottom face} = EA + EA = 2EA$
The total charge enclosed by the surface is $Q_{\text{enclosed}} = \sigma A$.
Applying Gauss’s law:
$\Phi_E = \dfrac{Q_{\text{enclosed}}}{\varepsilon_0} \implies 2EA = \dfrac{\sigma A}{\varepsilon_0}$
Which simplifies to:
$E = \dfrac{\sigma}{2\varepsilon_0}$
This result shows the electric field is constant in magnitude and always perpendicular to the plane sheet, regardless of the distance from the sheet.
A detailed explanation of Gauss's law and its application can be found in Electric Flux and Area Vector.
Formula Summary and Units
For a non-conducting infinite plane sheet of charge, the electric field at any point near the sheet is given by the following formula.
| Quantity | Value |
|---|---|
| Electric Field Magnitude ($E$) | $\dfrac{\sigma}{2\varepsilon_0}$ |
| Surface Charge Density ($\sigma$) | C/m$^2$ |
| Permittivity of Free Space ($\varepsilon_0$) | $8.85 \times 10^{-12}$ C$^2$/N·m$^2$ |
| SI Unit of Electric Field ($E$) | N/C or V/m |
The field’s direction is perpendicular to the sheet, and for a positively charged sheet, it points outward. For more details on electric field properties, see Electric Field Lines and Their Properties.
Electric Field for Conducting, Non-Conducting Sheets, and Parallel Sheets
In the case of a non-conducting infinite sheet, the field is present on both sides and is given by $E = \dfrac{\sigma}{2\varepsilon_0}$. For an infinite conducting sheet, all charge resides on one side, and the external electric field is $E = \dfrac{\sigma}{\varepsilon_0}$, while inside the conductor, the field is zero.
For two parallel sheets carrying equal but opposite surface charge densities, the resultant field between the sheets becomes $E = \dfrac{\sigma}{\varepsilon_0}$, and the fields outside cancel each other, making the net field zero outside.
Properties of the Electric Field Due to an Infinite Plane Sheet
- Field is uniform in magnitude and direction
- Field is independent of distance from the sheet
- Direction is perpendicular to sheet’s surface
- Field lines are parallel and evenly spaced
The symmetry ensures these properties hold at every location near the infinite sheet.
Application: Numerical Example
If an infinite non-conducting sheet has a surface charge density of $3 \times 10^{-6}$ C/m$^2$, the electric field at any point near the sheet is:
$E = \dfrac{\sigma}{2\varepsilon_0} = \dfrac{3 \times 10^{-6}}{2 \times 8.85 \times 10^{-12}} = 1.7 \times 10^{5}$ N/C$.
Sign does not affect the magnitude; use sign to determine the field direction.
Key Features and Physical Insights
- Field does not decrease with increasing distance
- Uniform at all points near the sheet
- Direction reverses for negative surface charge
- Commonly modeled in capacitor analysis
The assumption of infinite extent is idealized, but for practical calculations, sheets much larger than the observation distance are treated as infinite.
This concept is essential for understanding capacitors and charge distributions, as discussed in Electric Field Due to Infinite Plane.
Revision: Summary of Key Formulae
For a non-conducting infinite plane sheet, $E = \dfrac{\sigma}{2\varepsilon_0}$ on both sides. For an infinite conducting sheet, $E = \dfrac{\sigma}{\varepsilon_0}$ is present only on the outer side. For two parallel infinite sheets, the net field between them is $E = \dfrac{\sigma}{\varepsilon_0}$.
A thorough study of electric field intensity and similar derivations is available in Understanding Electric Field Intensity.
For complete revision of electrostatics topics relevant to JEE Main, visit Electrostatics Revision Notes.
FAQs on Understanding the Electric Field of an Infinite Plane
1. What is the electric field due to an infinite plane sheet of charge?
The electric field due to an infinite plane sheet of charge is uniform and does not depend on the distance from the sheet.
Key features:
- Magnitude: E = σ / (2ε0) where σ is surface charge density and ε0 is the permittivity of free space
- Direction: Perpendicular to the plane and away from the sheet (for positive charge)
- Uniformity: The field strength is the same at all points equidistant from the plane
2. Derive the expression for the electric field near an infinite plane sheet using Gauss's law.
The electric field near an infinite plane sheet is derived using Gauss's law and a Gaussian pillbox.
Follow these steps:
- Consider a Gaussian pillbox perpendicular to the sheet, with faces on both sides
- Net flux = E × 2A (through both faces), where A is face area
- Total charge enclosed = σ × A
- Apply Gauss's law: E × 2A = (σ × A) / ε0
- So, E = σ / (2ε0)
3. Why is the electric field of an infinite plane sheet independent of distance?
The electric field is independent of distance because field lines from an infinite plane are parallel and evenly spaced.
- Infinite extension means the density of field lines remains constant everywhere
- No edge effects are present to affect field strength
- Field does not weaken with distance, unlike point or line charges
4. What is the direction of the electric field produced by an infinite sheet of positive charge?
The electric field produced by an infinite positively charged sheet points perpendicularly away from the surface on both sides.
- For a positive sheet: Field is outward (away from the sheet)
- For a negative sheet: Field is inward (towards the sheet)
- Direction is always normal to the plane
5. How does the electric field due to two parallel infinite sheets of charge behave?
The net electric field between two parallel infinite sheets is the algebraic sum of each sheet's field.
Case analysis:
- If sheets have equal and opposite charges (σ and -σ): Field between them = σ/ε0
- Outside both sheets: Net field = zero
- Direction determined using sign of charges
6. What is surface charge density (σ) and how is it used in electric field calculations?
Surface charge density (σ) is the amount of electric charge per unit area spread over a surface.
- Formula: σ = q / A, where q is total charge and A is area
- Unit: C/m2
- Used in the formula for electric field: E = σ / (2ε0)
7. State Gauss's law and explain its application in finding the field due to an infinite plane.
Gauss's law states that the total electric flux through a closed surface is equal to the net charge enclosed divided by ε0.
- Mathematically: ∮E·dA = qencl/ε0
- Applied by choosing a symmetric Gaussian surface (like a pillbox) around the infinite sheet
- This approach simplifies evaluation of the uniform field E = σ / (2ε0)
8. What are the practical applications of the electric field due to an infinite plane sheet?
The concept of a uniform electric field from an infinite plane is used to model real-life parallel plate capacitors and shielding applications.
- Design of capacitors (used in electronics)
- Electrostatic shielding in laboratories
- Understanding field distribution in conductors and insulators
9. What happens to the electric field if the plane sheet is finite instead of infinite?
If the sheet is finite, the electric field is no longer perfectly uniform and decreases with distance.
- Edge effects become significant near the boundaries
- Far from the center, the field strength drops like a point charge
- The infinite sheet approximation is valid only near the center and for very large sheets compared to the observation distance
10. Does the electric field exist inside a conductor with an infinite plane of charge?
Inside a conductor, the static electric field is zero, even if the surface carries an infinite plane charge.
- Charges reside only on the surface of conductors
- Any internal field would be neutralized by free electron movement
- This illustrates electrostatic equilibrium in conductors, a common CBSE board concept
