Orthogonal Trajectories

In electromagnetism, electric field lines—representing the direction of the electric force—are always perpendicular to equipotential lines, along which potential energy remains constant.

These two sets of lines intersect at right angles, forming orthogonal trajectories.

To understand the concept mathematically, consider a family of parabolas opening along the x-axis, written as x = ky², where k is a constant.

Differentiating the equation with respect to x gives the slope of the tangent at any point on a parabola.

To generalize the result for all members of the family, eliminating k gives a slope expressed in terms of x and y.

The perpendicular direction to this slope is given by the negative reciprocal.

Substituting y/2x in the slope expression and rearranging gives the equation for the orthogonal trajectories of the parabolas

Integrating this equation gives a general expression that shows a family of ellipses centered at the origin.

These ellipses intersect the parabolas at right angles, mirroring the way electric field lines intersect equipotential surfaces in real physical systems.