Show that point charges \(+Q\) situated at \((\pm a, 0)\) in the \(x y\) plane give an E-field at large distances the same as that of a charge and a positive quadrupole at the origin. Hence argue that an atomic nucleus in the shape of a spheroid elongated in the \(x\) direction (prolate) would possess a positive quadrupole moment, that one contracted in the x direction (oblate) would possess a negative quadrupole moment and that neither would possess an electric dipole moment.

The electric field can be visualized as a vector field that surrounds electric charges. It exerts a force on other charges within the field's reach. An essential characteristic of the electric field is that its strength and direction are representative of the force that would be exerted on a positive test charge placed within the field. In mathematical terms, the electric field due to a point charge is given by the formula
\( E = \frac{kq}{r^2} \)
where \( E \) is the electric field, \( k \) is the electrostatic constant, \( q \) is the charge producing the field, and \( r \) is the distance from the charge to the point where the field is being measured. The electric field's direction is radially outward from positive charges and radially inward toward negative charges.

Point charges are idealized charges that are assumed to exist at a single, infinitesimally small point in space. They serve as a fundamental concept in electrostatics to simplify the study of electric fields, forces, and potentials. When point charges are positioned in space, they create an electric field in the surrounding area. The superposition principle states that if multiple point charges are present, the resultant electric field at any point is the vector sum of the electric fields due to each individual charge. This principle was used in the textbook exercise to calculate the electric field due to two point charges located symmetrically on the xy-plane:
\( E_{total} = E_1 + E_2 \).
Each field component, \( E_1 \) and \( E_2 \) for charges at positions \( (\pm a, 0) \) respectively, is calculated independently before being added together to find the total field.

The electric dipole moment is a vector quantity that provides a measure of the separation of positive and negative electrical charges within a system. For a dipole consisting of two charges of equal magnitude and opposite sign separated by a distance \( d \) the electric dipole moment \( \vec{p} \) is given by:
\( \vec{p} = q \cdot \vec{d} \)
where \( q \) is the magnitude of the charge and \( \vec{d} \) is the displacement vector pointing from the negative to the positive charge. The dipole moment is crucial in understanding the electric field produced by the dipole at large distances and plays a significant role in molecular and atomic physics. Unlike a system with a quadrupole moment, a system must have an asymmetry in the charge distribution to exhibit a dipole moment.

The atomic nucleus is the small, dense region at the center of an atom, where protons and neutrons, collectively called nucleons, are bound together by the strong nuclear force. The distribution of charges within the nucleus can be symmetric or asymmetric, impacting its electric properties. When the nucleus is elongated along one axis, like in a prolate spheroid, or flattened, like in an oblate spheroid, it exhibits a quadrupole moment. The quadrupole moment is a measure of the second-order deviation from spherical symmetry of a charge distribution. A positive quadrupole moment indicates a prolate distribution, while a negative one indicates an oblate distribution. In the case of these spheroids, no electric dipole moment exists because the charge distribution is symmetric with respect to the center, canceling out any potential dipole effect.