Show that the electric field at a point \(45^{\circ}\) from a dipole's axis is \(1 / \sqrt{5 / 8}\) times the field on the axis, assuming both points are the same distance from the dipole and that distance is large compared with the dipole spacing.

An electric dipole is a simple and fundamental system in the study of electromagnetism, consisting of two charges of equal magnitude but opposite sign, separated by a small distance. The dipole moment, symbolized by the vector \(\mathbf{p}\), is defined as the product of the charge magnitude (\(q\)) and the separation distance (\(d\)), and it points from the negative to the positive charge.

When considering electric fields generated by a dipole, the behavior of the field lines is critical to understand. These lines emanate from the positive charge and end at the negative charge, illustrating the direction in which a positive test charge would move in the presence of the dipole. The strength and direction of the field vary depending on the point of observation relative to the dipole's position, which brings into play the necessity of calculating the electric field at different angles and distances.

To calculate the electric field of a dipole (\(\mathbf{E}\)), one commonly used formula takes into account the angle (\(\theta\)) between the point of interest and the dipole axis, as well as the distance (\(r\)) from the dipole. This formula is often given as:\[E = \frac{k \cdot p \cdot (1 + 3cos^2(\theta))}{r^3}\]where \(k\) is Coulomb's constant, \(p\) is the magnitude of the dipole moment, and \(\theta\) is the angle away from the dipole's axis.

The textbook exercise challenges students to apply this formula at two different points: directly on the dipole's axis (where \(\theta=0\)) and at a \(45^\circ\) angle from the axis. By calculating and comparing the electric field values at these two points, one develops an understanding of how the dipole's field strength and distribution changes with angle, cementing the idea that the electric field is not uniform but varies depending on orientation and distance from the dipole.

The electric field of a dipole is vector in nature and can therefore be broken down into components based on the coordinate system in use. For a point in space at an angle \(\theta\) from the dipole’s axis, the electric field components can be resolved into parallel (\(E_{\parallel}\)) and perpendicular (\(E_{\perp}\)) components to the axis.

The perpendicular component is given by:\[E_{\perp} = \frac{k \cdot p \cdot sin(\theta)}{r^3}\]and the parallel component by:\[E_{\parallel} = \frac{k \cdot p \cdot (2 \cdot cos(\theta))}{r^3}\]Using trigonometry and the angle of interest, one can calculate these components and determine the total electric field vector at any point in the field created by the dipole. This approach shows the versatility of vector analysis in physics and helps students visualize and understand the multidimensional aspects of electric fields around dipoles.