Point charges \(+Q\) and \(-Q\) are separated by a distance \(2 l\), and \(O\) is the point midway between them. Find the E-field at \(P\), a distance \(r\) from \(O\), (a) when OP is along the line joining the charges and \(r>I\) and (b) when OP is perpendicular to the line joining the charges. Show that when \(r D\) l both fields are nearly proportional to \(Q l / r^{3}\) - (The charges form a dipole and \(\mathrm{P}\) is in the end-on and broadside positions, respectively -see Sec. 3.9.).

The concept of Electric charge is integral to understanding electric fields. Charges come in two types: positive and negative. Like charges repel each other, while opposite charges attract, as described by Coulomb's Law. In the context of dipoles, we typically deal with a pair of equal and opposite charges, denoted by \(+Q\) and \(-Q\), which are separated by a distance. Understanding these interactions allows us to predict the behavior of charged particles in electric fields, including the fields produced by dipoles.

Next, let's focus on the Electric field. It is a vector field that surrounds electric charges and represents the force that other charges would experience if placed in this field. The electric field (\(E\)) at a point in space is defined as the force (\(F\)) per unit charge (\(q\)) at that point, and it points in the direction of the force that a positive charge would experience. The formula for electric field intensity due to a point charge is:

Electric Field Intensity

Here, \(Q\) is the charge creating the field, \(r\) is the distance from the charge to the point of interest, and \(k\) remains the Coulomb's constant.

The Principle of Superposition is particularly relevant when calculating the net electric field from multiple charges, as in the case of dipoles. This principle says that the total electric field created by multiple charges is the vector sum of the electric fields produced by each charge independently. Therefore, when dealing with dipoles, we calculate the electric field due to the positive charge and the negative charge separately and then use vector addition to find the resultant electric field at a point in space.

A Dipole consists of two equal and opposite charges separated by a distance. The dipole's electric field is unique; it has a particular pattern that differs from the field of a single charge. The strength and direction of the electric field of a dipole vary with position. By analyzing two key positions, end-on (line joining the charges) and broadside (perpendicular to the line joining the charges), we can gain insights into the characteristics of the dipole's electric field.

Finally, understanding the Magnitude and Direction of the Electric field around a dipole is key to many applications in physics. The steps outlined in the solution showcase how to calculate both the magnitude and the direction. For instance, in end-on position (case a), the net electric field is along the dipole axis, whereas in the broadside position (case b), the net electric field is perpendicular to the dipole axis. Additionally, the magnitude of the electric field in either case can be shown to be proportional to \(Ql/r^3\) when the point of interest is much farther from the dipole than the separation between the charges (i.e., when \(r >> l\)). This inverse cubic relationship is pivotal in understanding how the electric field behaves at distances far from the dipole and is often utilized in approximations for more complex systems.