An electric dipole consists of two electric charges of equal...

An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are $q$ and $-q$ and are located at a distance $d$ from each other, then the electric field $E$ at the point $P$ in the figure is $$E=\frac{q}{D^{2}}-\frac{q}{(D+d)^{2}}$$ By expanding this expression for $E$ as a series in powers of $d / D,$ show that $E$ is approximately proportional to 1$/ D^{3}$ when $P$ is far away from the dipole.

An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are $q$ and $-q$ and are located at a distance $d$ from each other, then the electric field $E$ at the point $P$ in the figure is
$$E=\frac{q}{D^{2}}-\frac{q}{(D+d)^{2}}$$
By expanding this expression for $E$ as a series in powers of $d / D,$ show that $E$ is approximately proportional to 1$/ D^{3}$ when $P$ is far away from the dipole.

Key Concepts

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a finite distance. It is a key concept in electromagnetism, providing a simplified model to analyze how separated charges produce an electric field, especially in systems where the positive and negative charges are spatially displaced.

Dipole Moment

The dipole moment is a vector quantity defined as the product of the magnitude of one of the charges and the distance separating the charges. It characterizes the strength and orientation of a dipole, fundamentally influencing the resulting distribution of the electric field in space.

Electric Field

The electric field is a vector field that represents the force per unit charge exerted on a small test charge placed in the vicinity of other charges. In the context of a dipole, the field arises from the combined contributions of both charges and exhibits a distinct spatial dependence that can be derived using principles of vector superposition.

Series Expansion

Series expansion, such as a Taylor or binomial expansion, is a mathematical technique used to approximate functions by expressing them as a power series in a small parameter. In many physical contexts, including the analysis of the dipole field at large distances, this method simplifies complex expressions such that terms can be grouped according to their order of magnitude.

Far-Field Approximation

The far-field approximation involves analyzing the behavior of a field at distances large compared to the size of the source, in this case, the dipole separation. Under this approximation, higher-order terms in the series expansion become negligible, and the leading term—often responsible for the dominant behavior (here, 1/D^3 decay of the dipole's electric field)—provides the essential physical insight.

Please give Ace some feedback