What is the magnification of a marble?

James Tytko took on this question with the help of Richard Bowman from the University of Glasgow...

Richard - Thanks James. A clear marble will act as a lens – because the glass slows down the light that travels through it, it will bend light whenever it passes through the surface at an angle. This is known as refraction.

James - To calculate the optical magnification effect of the marble, we need to find out the focal length of the lens. This is the distance at which the light passing through the marble converges to a single point.

Richard - For that, we need to consider the radius of curvature of the marble and its refractive index (which measures how much light gets slowed down by the glass.) A typical marble made of normal glass will slow light down by about a factor of 1.5, and it might have a diameter of perhaps 12mm.

James - If you carefully trace the rays of light as they get bent on the way into and out of the marble, you come up with a focal length that’s a function of both the refractive index and the size, and for the numbers Richard mentioned that gives an effective focal length of 9mm. That means that you could hold the marble next to your eye, and be able to focus on an object only 9mm away.

Richard - The “magnification” this gives you is a surprisingly tricky concept: technically, the ratio of the marble’s focal length to the focal length of your eye sets how much bigger the image on your retina is than the object. However, this isn’t how we normally define magnification: you’re used to looking at objects quite a long way away, and most microscopes define magnification relative to looking at an object about 25cm from your eye (as you might do when reading).

James - That means the magnification will be the ratio of the focal length (9mm) to this distance (250mm) which suggests you have a magnification of about 28x – enough that you could easily see the pixels in a typical screen. If you use a smaller marble it scales linearly – so a 6mm marble would give you twice the magnification.

Richard - You’re right that scientists have known this for centuries – indeed the first microscopes were just small glass balls, and by using small ones (only a few millimetres) early scientists were already able to see microscopic features like cells: indeed the name “cell” came from the appearance of plant cells, which the observer thought resembled the cells in a monastery.

James - What a wonderful piece of trivia. Many thanks to the second to Nun, Richard Bowman from the University of Glasgow.

Thanks to Richard Bowman for the answer!