Sections#

Review & Summary#

Real and Virtual Images#

An image is a reproduction of an object via light. If the image can form on a surface, it is a real image and can exist even if no observer is present. If the image requires the visual system of an observer, it is a virtual image.

Image Formation#

Spherical mirrors, spherical refracting surfaces, and thin lenses form images by redirecting rays from the object. The image occurs where redirected rays cross (real) or where backward extensions cross (virtual). For rays close to the central axis, with object distance \(p\) (positive) and image distance \(i\) (positive for real, negative for virtual):

1. Spherical mirror:

(248)#\[ \frac{1}{p} + \frac{1}{i} = \frac{2}{r} = \frac{1}{f} \]

where \(f\) is the focal length and \(r\) is the radius of curvature. A plane mirror has \(r \to \infty\), so \(p = -i\). Real images form on the object side; virtual on the opposite side.

2. Spherical refracting surface:

(249)#\[ \frac{n_1}{p} + \frac{n_2}{i} = \frac{n_2 - n_1}{r} \]

where \(n_1\) and \(n_2\) are the indices of refraction. Convex surface facing object: \(r > 0\); concave: \(r < 0\).

3. Thin lens:

(250)#\[ \frac{1}{p} + \frac{1}{i} = \frac{1}{f} = (n-1)\left(\frac{1}{r_1} - \frac{1}{r_2}\right) \]

where \(n\) is the lens index and \(r_1\), \(r_2\) are the radii of the two surfaces. Real images form on the side opposite the object; virtual on the same side.

Lateral Magnification#

The lateral magnification \(m\) produced by a spherical mirror or thin lens is

(251)#\[ m = -\frac{i}{p} = \frac{h'}{h} \]

where \(h\) and \(h'\) are the heights of the object and image (perpendicular to the central axis). The image is upright if \(m > 0\), inverted if \(m < 0\).

Optical Instruments#

Simple magnifier: angular magnification \(m_\theta = 25\,\mathrm{cm}/f\), where \(f\) is the focal length and 25 cm is the near-point distance. Compound microscope: overall magnification \(M = m \cdot m_\theta\) with objective and eyepiece. Refracting telescope: angular magnification \(m_\theta = -f_{\mathrm{ob}}/f_{\mathrm{ey}}\).