Irodov Problem 1.241

Problem Statement

Find the gravitational field on the axis of a thin ring of mass M, radius R, at distance x from the center.

Given

Ring: M, R. Point on axis at distance x.

Concepts & Formulas

Only axial components survive by symmetry. Each element contributes dg·cosθ along axis, where cosθ = x/√(x²+R²).

Step-by-Step Solution

Step 1: Distance from ring element to point: l = √(x²+R²).
Step 2: dg_axial = G·dm/l² · (x/l) = G·dm·x/(x²+R²)^{3/2}.
Step 3: g = GMx/(x²+R²)^{3/2}.
Step 4: Maximum at x = R/√2: g_max = 2GM/(3√3·R²).

Worked Calculation

g = GMx/(x²+R²)^{3/2}.

Boxed Answer

g = GMx/(x² + R²)^{3/2}

Physical Interpretation

The field peaks at x = R/√2 and falls to zero at center (x=0) and at infinity. Helmholtz coils use this ring-field shape to create near-uniform magnetic fields.