APPLICATIONS OF THE ELLIPTIC INTEGRALS IN EVALUATIONS OF THE GRAVITATIONAL POTENTIAL AND FIELD INTENSITY OF A RING WITH ITS MASS UNIFORMLY DISTRIBUTED

Gravitational and electrostatic fields are two important types of conservative fields, and the analytical evaluation of their potentials and field intensities plays an important role in college physics teaching and practical applications. Based on the superposition principle, this paper evaluates the gravitational potential and field intensity at arbitrary points in space for a thin ring with uniformly distributed mass by using complete elliptic integrals and the derivative relations of elliptic integrals with respect to the modulus. Analytical expressions are obtained, and approximate results are derived for three special cases: when the field point lies on the axis of the ring, when it is far from the ring, and when it is close to the ring. As a verification and extension of the method, the electric potential and electric field intensity of a uniformly charged thin circular ring are further evaluated, and corresponding analytical expressions are obtained. The precautions required for the correct application of these results are also discussed. Finally, pedagogical suggestions are given on how to introduce elliptic integrals and how to teach the evaluation of the potential and field intensity at off-axis field points in college physics. A suggested teaching schedule is presented, the physical meaning of the modulus is discussed, and the distributions of the gravitational potential for the cases r≪a, r≈a, and r≫a are visualized using Matlab.