Two charges, \(+e\) and \(-e,\) are a distance of \(0.68 \mathrm{nm}\) apart in an electric field, \(E,\) that has a magnitude of \(4.4 \mathrm{kN} / \mathrm{C}\) and is directed at an angle of \(45^{\circ}\) with the dipole axis. Calculate the dipole moment and thus the torque on the dipole in the electric field.

Electric charges are one of the fundamental properties of matter, existing in two types: positive and negative. Like charges repel each other, while opposite charges attract. The unit of charge is the coulomb (C), and the charge of a single electron (or proton) is approximately \[\begin{equation}-1.6 \times 10^{-19} \text{C (electron)}, \ 1.6 \times 10^{-19} \text{C (proton)}. \end{equation}\] The interactions between charges are described by Coulomb's law, which states the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This concept is crucial when analyzing the behavior of electric dipoles in an external electric field, as seen in the exercise provided.

An electric field is a region around electric charges in which other charges experience a force. The strength and direction of this field are essential as they dictate how a charge will react when placed within this field. The electric field strength is measured in newtons per coulomb (N/C). An important property of electric fields is that they are vector fields, meaning they have both magnitude and direction, which was vital in solving the given exercise. The effect of an electric field on a dipole depends on its orientation relative to the field, hence, when the field is angled with respect to the dipole axis, the field's components must be considered separately to understand the total effect on the dipole.

A torque is a measure of the force that causes an object to rotate. When an electric dipole, which consists of two equal but opposite charges separated by some distance, is placed in an electric field, the field exerts a torque on the dipole. This torque depends on the dipole moment, which is a vector quantity defined as the product of the charge magnitude and the distance between the charges, oriented from the negative to the positive charge. Mathematically, the torque on a dipole is given by the cross product of the dipole moment (\[\begin{equation}\vec{p}\end{equation}\]) and the electric field (\[\begin{equation}\vec{E}\end{equation}\]). Therefore, the resulting torque tends to align the dipole with the electric field, which is a key concept in understanding the exercise and determining the magnitude and direction of the torque that the dipole experiences in the given scenario.