Show that the equations of the E-lines of a dipole are \(\left(\sin ^{2} \theta\right) / r=\) constant and of cquipotentials are \((\cos \theta) / r^{2}=\) constant.

Imagine standing in a gently flowing river where the water's motion represents the electric field created by a dipole. The river's flow lines, which curve and bend around obstacles, resemble the electric field lines that emanate from one charge of the dipole and merge into the other. These lines offer a visual guide to the direction of the electric field at any given point.

For an electric dipole, field lines emerge perpendicularly from the positive charge and curve as they approach the negative charge, where they terminate perpendicularly as well. The equation \(\left(\sin^{2}\theta\right)/r=\) constant, as derived in the exercise, captures the unique shape of these lines around a dipole. The intensity decreases with distance, illustrated by the lines spreading apart, signifying a weaker field at greater distances from the dipole.

Now, let's shift our focus to the tranquil, horizontal layers of the river, which remain at the same depth, no matter how the bottom shape twists below. These layers are akin to the equipotential surfaces surrounding the dipole, representing regions where the electric potential, or the 'electric height,' remains constant.

They're like contour lines on a map that mark points of equal altitude. You don't gain or lose energy moving along these lines, much like a hiker wouldn’t climbing along a contour. For a dipole, these surfaces are perpendicular to the electric field lines. The equation \(\left(\cos\theta\right)/r^{2}=\) constant showcases the spatial relationship between the potential and the position in the field around the dipole. These surfaces help in understanding how a charge would behave when placed in the dipole's electric field—it would feel a force urging it to move from regions of high potential to low potential, but on an equipotential surface, it would move freely without gaining energy.

If the river had floating markers aligning with the flow—the strength and direction of the current could be represented by arrows, with both magnitude and direction. These markers are analogous to the dipole moment, a vector quantity symbolized by \(\mathbf{p}\) that encapsulates an electric dipole's strength and orientation.

The dipole moment points from the negative charge to the positive charge and its magnitude is the product of the charge magnitude \(\left(q\right)\) and the separation distance \(\left(d\right)\), leading to \(\mathbf{p}=q\times d\). It is pivotal when calculating the electric field and potential created by the dipole, serving as a key component in the equations we've examined. The dipole moment directly influences a charge's interaction with the dipole's electric field, and thus, it plays a crucial role in understanding the overall behavior of dipoles in an electric field.