2.7 The simple magnifier  (Page 2/4)

From part (b) of the figure, we see that the absolute value of the image distance is $\left|d_{\text{i}}\right|=L-\mathcal{\ell }$ . Note that $d_{\text{i}}<0$ because the image is virtual, so we can dispense with the absolute value by explicitly inserting the minus sign: $-d_{\text{i}}=L-\mathcal{\ell }$ . Inserting this into [link] gives us the final equation for the angular magnification of a magnifying lens:

Note that all the quantities in this equation have to be expressed in centimeters. Often, we want the image to be at the near-point distance ( $L=25\text{cm}$ ) to get maximum magnification, and we hold the magnifying lens close to the eye ( $\mathcal{\ell }=0$ ). In this case, [link] gives

which shows that the greatest magnification occurs for the lens with the shortest focal length. In addition, when the image is at the near-point distance and the lens is held close to the eye $(\mathcal{\ell }=0)$ , then $L=d_{\text{i}}=25\text{cm}$ and [link] becomes

where m is the linear magnification ( [link] ) derived for spherical mirrors and thin lenses. Another useful situation is when the image is at infinity $(L=\text{∞})$ . [link] then takes the form

The resulting magnification is simply the ratio of the near-point distance to the focal length of the magnifying lens, so a lens with a shorter focal length gives a stronger magnification. Although this magnification is smaller by 1 than the magnification obtained with the image at the near point, it provides for the most comfortable viewing conditions, because the eye is relaxed when viewing a distant object.

By comparing [link] with [link] , we see that the range of angular magnification of a given converging lens is

Summary

• A simple magnifier is a converging lens and produces a magnified virtual image of an object located within the focal length of the lens.
• Angular magnification accounts for magnification of an image created by a magnifier. It is equal to the ratio of the angle subtended by the image to that subtended by the object when the object is observed by the unaided eye.
• Angular magnification is greater for magnifying lenses with smaller focal lengths.
• Simple magnifiers can produce as great as tenfold ( $10\times$ ) magnification.

Problems