The Simple Microscope

A lens that lets the object come closer

The first lesson ended with a limit. An object looks larger the nearer it comes, but only down to the near point, \(25\,\text{cm}\) away. A convex lens held close to the eye removes that limit. Place the object inside the focus of the lens. The lens then forms a virtual, erect, enlarged image on the same side as the object. The eye looks at this image, not at the object.

The image can be placed where the eye is comfortable. Anywhere from \(25\,\text{cm}\) to infinity is within the eye's range. A convex lens used in this way is a simple microscope, or magnifying glass. The object must lie inside \(f\), because only there does a convex lens give a virtual, erect image on the object's side. An object beyond \(f\) would give a real, inverted image behind the eye.

In the first chip of the figure above, the letter, \(1\,\text{cm}\) tall, stands at \(25\,\text{cm}\) with no lens. It makes the angle \(\alpha = 1/25 = 0.040\,\text{rad}\). This is the best the unaided eye can do. In the second chip the lens is in place and the letter stands at \(4.2\,\text{cm}\). Its image is at \(25\,\text{cm}\), six times taller, and the angle is \(0.24\,\text{rad}\). In the third chip the letter is at the focus. The rays leave parallel, the image is at infinity, and the angle is \(0.20\,\text{rad}\).

Magnifying power of a simple microscope

Hold the lens close to the eye. Then the ray from the top of the object through the centre of the lens goes straight into the eye. The angle at the eye is \(\beta = h/|u|\), where \(|u|\) is the distance of the object from the lens. Without the lens, at the near point, the angle is \(\alpha = h/D\). So the magnifying power is \[ M = \dfrac{\beta}{\alpha} = \dfrac{h/|u|}{h/D} = \dfrac{D}{|u|}. \]

The object cannot come as close as we like. Its image must stay within the eye's range, from \(D\) to infinity. The two ends of that range give the two standard results. The nearest object puts its image at \(D\). The farthest, at the focus, puts it at infinity.

With the image at infinity the object sits at the focus, \(|u| = f\), and \(M = D/f\). Relaxing the eye costs exactly one unit of magnifying power. For image at \(D\) the linear magnification is also \(m = v/u = -25/-4.17 = 6\), equal to \(M\). For the image at infinity, \(m\) has no meaning, but \(M = 5\) does.

For \(f = 5\,\text{cm}\), the image at \(D\) gives \(M = 1 + 25/5 = 6\), and the object stands at \(|u| = 25 \times 5/30 = 4.17\,\text{cm}\). The image at infinity gives \(M = 25/5 = 5\). These are the numbers of the chips. The erect virtual image gives \(M > 0\) in both cases.

Why a single lens stops at about ten

The formula \(M = 1 + D/f\) seems to promise any magnification. A shorter focal length gives a larger \(M\). Fig 3.1 shows how steeply \(M\) rises as \(f\) falls below a few centimetres. For \(M = 100\), \(f\) would be only \(25/99 = 0.25\,\text{cm}\).

A short focal length needs strongly curved faces. An equiconvex glass lens of \(\mu = 1.5\) has \(f = R\), so a lens of \(f = 0.25\,\text{cm}\) needs faces of radius \(2.5\,\text{mm}\). Such a lens is tiny and must be held almost touching the eye. Worse, its rays are no longer paraxial. The chapter on lenses showed that edge rays then focus nearer (spherical aberration) and that colours focus apart (chromatic aberration). Both blur the image as the curvature grows.

In practice a single lens gives a clear image up to about ten times. Beyond that, a better instrument is needed. It uses two lenses, each doing part of the job, and the next lesson builds it.