An electric dipole on the x -axis consists of a charge -Q at...

An electric dipole on the $x$ -axis consists of a charge $-Q$ at $x=-d$ and a charge $+Q$ at $x=+d$. (a) Find a general expression (in terms of $Q$ and $d$ ) for the electric field at points on the $x$ -axis with $x>d$. (b) Show that for $x \gg d$ the magnitude of the field you computed in part (a) is a pproximately equal to $E=4 k Q d / x^{3}$ and therefore approximates an inverse-cube distance law.

Key Concepts

Far-Field Approximation

The far-field approximation is an analytical approach used when the distance from the dipole is much greater than the separation between the charges. In this regime, higher-order terms become negligible, and the electric field exhibits an inverse-cube distance dependence, simplifying the description of the dipole's effects at large distances.

Superposition Principle

The superposition principle is the concept that the total electric field due to multiple charges is the vector sum of the electric fields produced by each individual charge. This principle is crucial for calculating the electric field of systems like dipoles, where fields from positive and negative charges combine.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance. It is a fundamental concept in electromagnetism that illustrates how spatial separation of charges gives rise to a characteristic field pattern, which is distinct from that of a single point charge.

Dipole Moment

The dipole moment is a vector quantity that characterizes the strength and orientation of an electric dipole. It is defined as the product of one of the charges and the distance between the charges, and it plays a central role in determining the behavior of the dipole’s electric field.

Coulomb's Law

Coulomb's law provides the foundation for understanding the electric field generated by a point charge. It states that the magnitude of the electric field due to a point charge is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge, serving as the basis for calculating the contributions of each charge in a dipole.

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