Question

The electric field \(E\) is measured at a point \(P(0,0, d)\) generated due to various charge distributions and the dependence of \(E\) on \(d\) is found to be different for different charge distributions. List-I contains different relations between \(E\) and \(d\). List-II describes different electric charge distributions, along with their locations. Match the functions in List-I with the related charge distributions in List-II. LIST-I P. \(E\) is independent of \(d\) Q. \(E \propto \frac{1}{d}\) R. \(E \propto \frac{1}{d^{2}}\) S. \(E \propto \frac{1}{d^{3}}\) LIST-II 1\. A point charge \(Q\) at the origin 2\. A small dipole with point charges \(Q\) at \((0,0, l)\) and \(-Q\) at \((0,0,-l)\) Take \(2 l \ll d\) 3\. An infinite line charge coincident with the \(x\) -axis, with uniform linear charge density \(\lambda\) 4\. Two infinite wires carrying uniform linear charge density parallel to the \(x\) - axis. The one along \((y=0, z=l)\) has a charge density \(+\lambda\) and the one along \((y=0, z=-l)\) has a charge density \(-\lambda\). Take \(2 l \ll d\) 5\. Infinite plane charge coincident with the \(x y\) -plane with uniform surface charge density

Short answer

P matches with 5, Q matches with 3, R matches with 1, and S matches with both 2 and 4.

Step-by-step solution

Understanding the Physical Context

Firstly, understand how electric field intensity varies with the distance from different types of charge distributions. For a point charge, the electric field follows an inverse square relationship (Coulomb's law). For a small dipole, the electric field varies as the inverse cube of the distance. For an infinite line charge, the electric field varies inversely with the distance, and for an infinite plane charge, the electric field is constant and does not depend on the distance.

Matching P - Electric Field Independent of Distance

Since the electric field is independent of the distance for an infinite plane charge, match P with 5. This is because an infinite plane charge creates a uniform electric field.

Matching Q - Electric Field Proportional to 1/d

An infinite line charge creates an electric field that is inversely proportional to the distance from the line. Therefore, match Q with 3.

Matching R - Electric Field Proportional to 1/d^2

A point charge at the origin exhibits an electric field that follows the inverse square law. Hence, match R with 1.

Matching S - Electric Field Proportional to 1/d^3

For the small dipole, given the condition that the separation between charges is much smaller than the distance to the point P, the electric field strength decreases as the cube of the distance. Thus, match S with 2.

Assessment of the Two Infinite Wire System

The system of two infinite wires with opposite charge densities, separated by a distance much smaller than the distance to the point P, can be considered as a dipole. Its electric field would also decrease as the cube of the distance (1/d^3). Thus, option S also matches with 4.

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electromagnetism that describes the force between two point charges. According to this law, the magnitude of the electrostatic force between two stationary point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

Mathematically, it is expressed as: \[ F = k \frac{|q_1 q_2|}{r^2} \] where:\begin{itemize}\item \( F \) is the magnitude of the force.\item \( q_1 \) is the magnitude of the first charge.\item \( q_2 \) is the magnitude of the second charge.\item \( r \) is the distance between the charges.\item \( k \) is the Coulomb constant, approximately equal to \( 8.987 \times 10^9 N m^2/C^2 \).\end{itemize}
Coulomb's Law not only determines the strength of the force but also provides insight into the electric field created by a point charge, which is important for understanding charge distributions. The electric field (\( E \) produced by a point charge (\( Q \) can therefore be described as being proportional to the charge and inversely proportional to the square of the distance from the charge, leading to the relationship: \[ E \: \propto \frac{Q}{d^2} \] This inversely squared relationship with distance is a characteristic footprint of a point charge field, and it is critical when analyzing electric fields in various configurations.

Electric Field Intensity

Electric field intensity, often referred to simply as the electric field (\( E \)), is a vector quantity that represents the force experienced per unit positive charge at a given point in space. The direction of the field is the direction that a positive test charge would move if placed in the field.

The electric field is important because it is the means through which charges interact with one another even when they are not in direct contact. It conveys the influence of one charge on the space around it, which, in turn, affects other charges within that space. This concept bridges the importance of understanding electric fields when studying charge distributions, such as points, dipoles, lines, and planes.

The intensity or strength of an electric field created by a point charge reduces with distance, following the inverse square law linked to Coulomb's Law. However, this relationship changes when dealing with different types of charge distributions, such as lines or planes, which have their unique distance dependencies.

Charge Distribution Types

Electric charge distributions can be broadly categorized into several types based on their geometry, and this categorization influences the behavior of their respective electric fields. Here are some of the common types:

• Point Charges: A point charge is considered to be a charge located at a single point in space. Its field strength follows an inverse square dependency on distance.
• Dipoles: Consisting of two equally and oppositely charged point charges separated by a small distance, the field of a dipole decreases at a rate proportional to the inverse cube of the distance.
• Infinite Line Charges: A continuous distribution of charge along an infinitely long line exhibits a field that decreases inversely with distance (linearly).
• Uniformly Charged Infinite Planes: Infinite plane charges create a uniform electric field that remains constant in magnitude and does not vary with distance from the plane.

These diverse types of charge distributions demonstrate different spatial behaviors of electric fields, and are used to illustrate various concepts in electromagnetism, preparing students for more complex scenarios.

Electric Field Distance Dependence

The distance dependence of an electric field is a function of the spatial configuration of the charge distribution creating it. For electric fields, this dependence is not always intuitive, and each type of charge distribution follows its unique distance-related behavior:

• Point Charges: For point charges, the electric field diminishes with distance according to the inverse square law (\( E \: \propto \frac{1}{d^2} \)).
• Dipoles: The dipole's field diminishes more rapidly, following an inverse cube law (\( E \: \propto \frac{1}{d^3} \)), due to the vector cancellation of closely positioned opposite charges.
• Infinite Line Charges: An infinite line of charge diminishes linearly with distance (\( E \: \propto \frac{1}{d} \)), as the influence of the line extends uniformly across its length.
• Infinite Planes: In contrast, the field of an infinite plane charge remains constant regardless of the distance (\( E \: \propto d^0 \)), a surprising result given our experience with point charges.

This varied distance dependence highlights the need to understand the nature of the source of the electric field in order to predict the behavior of the field at various distances. It's an essential concept that helps students grasp how electric fields propagate through space and affect the surrounding charges.