Gravitation and the Principle of Superposition — Question 6

Question 6

A ring has zero field at its center but an axial maximum elsewhere

A uniform thin ring has mass $M$ and radius $R$. A test point lies a distance $z>0$ along its symmetry axis.

1. Derive the axial field by resolving the field of each mass element.

2. Find the value of $z$ where the field magnitude is greatest.

Question 6 - Solution

A ring has zero field at its center but an axial maximum elsewhere

(a) Derivation and calculation

All elements have distance $s=\sqrt{R^2+z^2}$ from the axial point. Their transverse contributions cancel: $$dg_z=-\frac{G,dm}{s^2}\frac zs, \qquad g_z=-\frac{Gz}{s^3}\int dm =\boxed{-\frac{GMz}{(R^2+z^2)^{3/2}}}.$$

(b) Result and interpretation

For $z>0$, differentiate the magnitude: $$\frac{d|g_z|}{dz}=GM\frac{R^2-2z^2}{(R^2+z^2)^{5/2}}.$$ It changes from positive to negative at $$\boxed{z=R/\sqrt2},\qquad \boxed{|g|_{\max}=\frac{2GM}{3\sqrt3R^2}}.$$ The field vanishes at the center and approaches $-GM/z^2$ far away.

Physical check

The interior axial field is restoring near $z=0$; the maximum refers to its magnitude.