Distinguish mirror image signs and the focal limit

Establish the model and evidence

A concave spherical mirror is converging in the paraxial approximation; a convex mirror is diverging. With the real-is-positive convention, concave focal length is positive and convex focal length negative. The signed radius of curvature is twice the focal length for this approximation. Rays far from the principal axis require attention to spherical aberration, which is not modeled here.

Use 1/f=1/o+1/i with a real object o>0. For mirrors, i>0 locates a real image in front of the mirror, while i<0 locates a virtual image behind it. The spatial diagram puts the mirror at x=0 and the object at negative x, so a real mirror image has x=−i. Do not confuse the sign convention for image distance with the sign of a global coordinate.

Magnification M=−i/o gives image height relative to object height. A real image of the real object in this simple mirror case is inverted; a virtual image is upright. A concave mirror can give either, depending on whether the object is outside or inside its focal distance. A convex mirror with a real object gives an upright reduced virtual image.

At o=f for a concave mirror, the algebraic denominator vanishes because reflected rays are parallel. The correct state is no finite image, often described as image at infinity. The browser does not store an infinite coordinate or pretend a finite screen can focus it. Two principal rays suffice to locate a finite image; a third can provide an independent construction check.

Predict a third principal ray before checking the image: for a concave mirror, an incident ray whose line passes through the focus reflects parallel to the axis. For a convex mirror, aim the incident ray toward the virtual focus behind the mirror; its reflected ray is parallel. Sketch this focal-directed rule alongside the parallel and vertex rays. Check that the third reflected ray or its backward extension passes through the same finite image tip. When the object is inside the concave focal distance, the incident line may need a backward extension through the focus; that construction is not light passing behind the object.

Worked evidence

For a concave mirror f=+10 cm and object o=30 cm, i=+15 cm and M=−0.5. At o=5 cm, i=−10 cm and M=+2. At o=10 cm, outgoing rays are parallel and there is no finite i.

Predict and investigate

As a concave-mirror object crosses the focal distance, does the image remain real on both sides?

Compare concave cases o=5, 10, 15 and 30 cm, then a convex case at 30 cm. Use the ray intersections or backward extensions to explain each sign and the exact focal boundary.

Transfer to changed conditions

A concave mirror has radius of curvature +40 cm and object distance 60 cm. Find f, i and M.

Compare the worked transfer

f=+20 cm, i=+30 cm and M=−0.5. Actual reflected rays meet in front of the mirror, producing a real inverted image at half the object height under the paraxial assumptions.

Explain the boundary

Why is an undefined finite image at o=f a physical model limit rather than a numerical result to replace with zero?

Sources and authorship

Physical relationship reference. Current AP Physics 2 framework, checked 30 October 2026. Original text, rays and computed/synthetic records, not completed optical measurements. All investigations are virtual: no lasers, intense-light viewing, sunlight focusing, flames or eye tests are requested. Any real practical work needs qualified supervision and independent safety review.