Problem 15 - Master Resource Book in JEE Main Physics

Question

In infinite parallel plane sheet of a metal is charged to charge density o coulomb per square metre in a medium of dielectric constant \(K\). Intensity of electric field near the metallic surface will be (a) \(E=\frac{\sigma}{\varepsilon_{0} K}\) (b) \(E=\frac{K}{3 \varepsilon_{0}}\) (c) \(E=\frac{\sigma}{2 \varepsilon_{0} K}\) (d) \(E=\frac{K}{2 \varepsilon_{0}}\)

Step-by-Step Solution

Answer

The electric field intensity near the surface is \(E = \frac{\sigma}{2\varepsilon_0 K}\), which corresponds to option (c).

Step 1: Understanding the Problem

We are given an infinite plane sheet with a charge density \(\sigma\) in a medium of dielectric constant \(K\). We need to find the electric field intensity (\(E\)) near the metallic surface.

Step 2: Reviewing the Formula for Electric Field

For an infinite charged plane without a dielectric medium, the electric field intensity is \(E = \frac{\sigma}{2\varepsilon_0}\), where \(\varepsilon_0\) is the permittivity of free space.

Step 3: Incorporating Dielectric Constant

The presence of a dielectric medium affects the electric field by a factor of \(K\), the dielectric constant. When a dielectric is present, the electric field is reduced by this factor, hence the formula becomes \(E = \frac{\sigma}{2\varepsilon_0 K}\).

Step 4: Verifying the Options

Compare the derived formula \(E = \frac{\sigma}{2\varepsilon_0 K}\) with the given options. The correct answer matches option (c).

Key Concepts

Dielectric Constant

Understanding the dielectric constant is crucial for comprehending electric fields in a medium. The dielectric constant, often denoted as \(K\), is a dimensionless number that indicates how much an electric field is reduced inside a medium compared to a vacuum. This reduction occurs because the dielectric material polarizes in response to the applied electric field, thereby opposing it.
When a dielectric medium is present, instead of having the electric field (\(E\)) in free space, you will have it divided by the dielectric constant \(K\). It's important because this division implies the medium significantly weakens the original field effect. Therefore, in mathematical terms, if you know the original field without the medium is given by \(E_0 = \frac{\sigma}{2\varepsilon_0}\), introducing the dielectric gives us \(E = \frac{\sigma}{2\varepsilon_0 K}\). This is vital when calculating electric fields near charged surfaces, especially metals in a dielectric medium.

Infinite Plane Sheet

The concept of an infinite plane sheet is a theoretical idealization. In practice, no plane sheet is truly infinite; however, considering it to be so simplifies calculations in electrostatics. An infinite plane sheet means the sheet extends without end, effectively removing the boundary effects that finite-sized sheets exhibit.
In electrostatics, when dealing with an infinite plane sheet with a uniform charge density \(\sigma\), the electric field intensity at any point close to the sheet can be described by the formula \(E = \frac{\sigma}{2\varepsilon_0}\), assuming no medium interference.
This setup is vital because unlike point charges whose field decreases with distance, an infinite plane surface creates a uniform field, making the study of its effects, such as on dielectrics or conductors, much easier. This infinite extension concept helps achieve meaningful solutions without concern for edge effects or directional variance.

Charge Density

Charge density is a fundamental quantity in electrostatics, denoted by \(\sigma\), representing the amount of electric charge per unit area on a surface. The units of charge density are typically coulombs per square meter (C/m²).
This measure is critical when calculating electric fields, as it directly impacts the intensity of the field produced. For example, an infinite, uniformly charged plane sheet employs \(\sigma\) in its computation for field intensity \(E\). A higher charge density, \(\sigma\), means a stronger electric field emanates from the charged surface.
In context to the problem we discussed, the charge density affects the electric field intensity according to the formula \(E = \frac{\sigma}{2\varepsilon_0 K}\), where the sheet's substantial charge distribution in the presence of a dielectric affects the magnitude of \(E\). Understanding charge density is, therefore, pivotal for predicting the behavior of electric fields in and around charged surfaces.