The derivation of the electric field due to a conducting sheet is one of the most important topics in class 12 physics, under electrostatics. Here we will also learn about concepts, formulas, applications, and exams-focused insights.
What is an Infinite Conducting Sheet?
An infinite plane sheet of charge is a surface extending endlessly in all directions with a uniform surface charge density. The electric field due to it is uniform and always perpendicular to the surface.
Definition of a conducting sheet
A conducting sheet is a surface made of conducting material in which charges are free to move on the entire surface. In electrostatic equilibrium, all excess charge resides on the surface, and the electric field inside the conductor is zero.
The electric field exists only just outside the surface and is perpendicular to it. Its magnitude is: `E = \frac{\sigma}{\epsilon_0}`.
Concepts Used in this Derivation
Electric Field in Electrostatics
The region around a charge where another charge experiences force is known as the electric field.
Surface Charge Density (`\sigma`)
Charge per unit area on a surface is known as the surface charge density. It is represented by `\sigma = Q/A,` and unit is `C/m^2.`
Conductors in Electrostatic Equilibrium
The charge given to a conductor resides only on its surface, and the free charges redistribute until the internal electric field becomes zero.
Direction of Electric Field Near Conductor
The electric field at the surface is always perpendicular to a conductor.
The electric field inside a Conductor is Zero
Inside a conductor, the electric field is zero because charges redistribute themselves on the surface, canceling any internal electric field.
Derivation of Electric Field near a Charged Infinite Conducting Sheet
Calculation of the Electric field
Consider a uniformly charged conducting sheet with surface charge density of `\sigma`.
We need to determine the electric field at a point P located near the conducting sheet.
The direction of the electric field intensity at point P is perpendicular to the surface of the conducting sheet.
To find the electric field near the sheet, we choose a cylindrical Gaussian surface (pillbox) such that:
- One circular face, `S_1` lies outside the conductor.
- The other circular face `S_2` lies inside the conductor.
- The third surface `S_3` is curved.
Total electric flux through the Gaussian surface is given by Gauss's law:
`\phi_E = \oint_s \vec E.d\vec A = \frac{q'}{\epsilon_0}`
The total flux can be written as the sum of the flux through all three surfaces:
`\int_{s_1} \vec E.d\vec A + \int_{s_2}\vecE.d\vecA + \int_{s_3}\vecE.d\vecA = \frac{q'}{\epsilon_0}`
For a conductor in electrostatic equilibrium, free charges rearrange themselves such that the internal electric field becomes zero (E = 0).
`\int_{s_1} E dA cos 0^\circ+ \int_{s_2} 0 dA cos\theta + \int_{s_3} E dA cos 90^\circ = \frac{\sigma A}{\epsilon_0}` `( q' = \sigma A)`
`\int_{s_1} E dA (1)+ \int_{s_2} 0 dA cos\theta + \int_{s_3} E dA (0) = \frac{\sigma A}{\epsilon_0}`
Simplifying,
`\int_{s_1} E dA (1)+ 0 + 0 = \frac{\sigma A}{\epsilon_0}`
`E \int_{s_1} dA + 0 + 0 = \frac{\sigma A}{\epsilon_0}`
`E \int_{s_1} dA = \frac{\sigma A}{\epsilon_0}`
`E A = \frac{\sigma A}{\epsilon_0}` `(\because \int_{s_1} dA = A)`
Thus, we get:
`E A= \frac{\sigma A }{\epsilon_0}`
Final Formula
`E = \frac{\sigma }{\epsilon_0}`
Here,
`\sigma = \text{Charge density}`
`\sigma = \frac{\text{Charge}}{\text{Area}}`
`q' = \text{Charge inside Gaussian Surface}`
In vector form
`\vec E = \frac{\sigma }{\epsilon_0} \hat n`
Result
Distance Independence
This formula shows that the electric field intensity near a conducting sheet does not change with distance; that is, the electric field around it remains uniform (constant).
Electric Field Outside the Sheet
The electric field just outside the conducting sheet is `E = \frac{\sigma}{\epsilon_0}`
The direction of electric field outside the conducting sheet is perpendicular to the surface.
Same magnitude on both sides of the conducting sheet.
The electric field does not depend on the distance from the sheet.
Electric Field Inside the Sheet
The electric field is zero inside the conducting sheet due to the rearrangement of charges.
Conclusion
- The electric field due to an infinite conducting sheet is derived using Gauss's law.
- Charge remains on the surface.
- The electric field inside the conductor is zero.
- The electric field just outside the sheet `E = \frac{\sigma}{\epsilon_0}.`
- Electric fields are independent of distance, uniform, and perpendicular to the surface.
- This topic is very important for exam concepts and numerical problems in electrostatics.
Real Life Application
- Parallel Plate Capacitors.
- Electrostatic Precipitators (Air Pollution Control).
- Electrostatic Shielding (Faraday Cage).
- Touch Screens and Sensors.
- Printing and Photocopiers (Xerography).
- To Create Uniform Field Region for Physics Experiments.
Questions and Answers
Q.1 What is Gauss’s Law in terms of total
electric flux through a closed surface in a vacuum of air?
Ans: According to Gauss’s Law, the total
electric flux through a closed surface is equal to the product of the net
charge inside the surface and ` \frac{1}{\epsilon_0}`.
Where,
`\phi = \frac{q}{\epsilon_0}`
q = Charge
inside the Gaussian surface
Q.2 What does `\Sigma q`
Ans : `\Sigma q` is the algebraic sum of the charges inside the closed surface.
Q.3 What is ` \epsilon_0` in Gauss’s Law?
Ans: ` \epsilon_0` is the permittivity of a vacuum.
Q.4 How is the constant K related to ` \epsilon_0` ?
Ans :
Q.5 What is the electric field due to a uniformly
charged conducting sheet with a surface charge density?
Ans : `E\ =\ \frac{\sigma}{\epsilon_0}`
Q.6 What is the direction of the electric field due to a uniformly charged conducting sheet?
Ans : The direction of the electric field due to
a uniformly charged conducting sheet is perpendicular to the surface of the sheet
and pointing away from positive charge.
Q.7 How does the electric field due to a
conducting sheet vary with distance?
Ans :
Independent
of the distance from the sheet. Remains constant.
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