A glass rod (n₂ = 1.50) in air (n₁ = 1.00) has a hemispherical end of radius R = +2.0 cm (convex toward the object). An object is placed 12 cm in front of the vertex. Find the image position and magnification. Step 1: Sign convention: u = −12 cm (left of vertex), R = +2.0 cm (centre to the right). Step 2: n₁/u + n₂/v = (n₂ − n₁)/R → 1/(−12) + 1.5/v = (1.5 − 1)/2.0 = 0.25. So 1.5/v = 0.25 + 1/12 = 0.25 + 0.0833 = 0.3333 → v = 1.5/0.3333 = +4.5 cm. Step 3: v positive → real image 4.5 cm inside the glass rod (to the right of the vertex). Magnification m = −(n₁v)/(n₂u) = −(1 × 4.5)/(1.5 × (−12)) = −4.5/(−18) = +0.25. Image is upright (for a single refracting surface, positive m means upright) and one-quarter size — the curved surface both focuses and minifies here.