Problem 18 - Master Resource Book in JEE Main Physics

Question

A point \(Q\) lies on the perpendicular bisector of an electrical dipole of dipole moment \(p\). If the distance of \(Q\) from the dipole is \(r\) (much larger than the size of the dipole), then the electric field intensity \(E\) at \(Q\) is proportional to (a) \(r^{-2}\) (b) \(r^{-4}\) (c) \(r^{-1}\) \((d) r^{-3}\)

Step-by-Step Solution

Answer

The electric field intensity \(E\) at point \(Q\) is proportional to \(r^{-3}\).

Step 1: Understanding the Electric Field due to a Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance. The dipole moment, denoted as \(p\), is the product of the charge and the separation. The electric field due to a dipole at a point on its perpendicular bisector is different compared to the field on its axial line.

Step 2: Using the Formula for Electric Field on the Perpendicular Bisector

For a point located on the perpendicular bisector of a dipole, the electric field intensity \(E\) is given by the formula: \[ E = \frac{k \cdot p}{r^3} \] where \(k\) is a constant and \(r\) is the distance from the dipole.

Step 3: Identifying the Proportional Relationship

From the formula \( E = \frac{k \cdot p}{r^3} \), it can be seen that the electric field intensity \(E\) is directly proportional to \(r^{-3}\). This means as the distance \(r\) increases, the electric field decreases with the cube of \(r\).

Key Concepts

Electric Field Intensity

Electric field intensity is a measure of the force experienced by a unit positive charge placed in an electric field. For an electric dipole, this intensity reflects how the presence of two close, opposite charges influences the space around them. When discussing electric field intensity, it's crucial to recognize that the dipole's effect greatly depends on the position relative to the dipole. The field varies in strength and direction, depending on whether it's measured on the axial line or the perpendicular bisector of the dipole. Ultimately, understanding electric field intensity helps predict how charges will behave in various field configurations.

Dipole Moment

The dipole moment is a vector quantity that represents the strength and direction of an electric dipole. It is defined as the product of the magnitude of one of the charges and the distance separating them. Mathematically, it is represented as: \( p = q imes d \) where \( q \) is the charge and \( d \) is the separation distance. Dipole moment gives us an intuitive sense of how strongly the dipole can affect its surrounding electric field. A larger dipole moment means a stronger influence, as the charges exert a greater force over a wider area. The direction of the dipole moment is conventionally from the negative to the positive charge.

Perpendicular Bisector

The perpendicular bisector of an electric dipole is a line that divides the dipole into two equal parts and stands at a right angle to the line joining the two charges. The field due to a dipole at the perpendicular bisector is distinct because, at any point on this line, the contributions to the electric field from both charges are equal in magnitude but opposite in direction. Thus, when calculating the electric field along this line, except directly at the midpoint, these opposing fields can have a net non-zero effect, pointing along the bisector itself. This symmetry makes the calculations more straightforward but distinct from other positions around the dipole, like the axial line.

Distance Dependency in Fields

The influence of an electric dipole on its surrounding space diminishes as the distance increases. This relationship depends on the position of the point in question relative to the dipole. On the perpendicular bisector, as described in the given exercise, the electric field intensity can be expressed as: \( E = \frac{k \cdot p}{r^3} \). Here, the electric field diminishes with the cube of the distance, meaning the intensity falls off very quickly as you move away from the dipole. Such a rapid decrease underscores the localized influence of electric dipoles, and explains why their effects become negligible at great distances.