Finding the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations

Since the object is placed farther away from a converging lens than the focal length of the lens, this situation is analogous to those illustrated in Figure 25.33 and Figure 25.34. Ray tracing to scale should produce similar results for $d_{\text{i}}$. Numerical solutions for $d_{\text{i}}$ and $m$ can be obtained using the thin lens equations, noting that $d_{\text{o}}=\mathrm{0.750\; m\; and}f=\mathrm{0.500\; m}$.

The ray tracing to scale in Figure 25.35 shows two rays from a point on the bulb’s filament crossing about 1.50 m on the far side of the lens. Thus the image distance $d_{\text{i}}$ is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus $m$ is about –2. The minus sign indicates that the image is inverted.

The thin lens equations can be used to find $d_{\text{i}}$ from the given information:

Rearranging to isolate $d_{\text{i}}$ gives

Entering known quantities gives a value for $1/d_{\text{i}}$:

This must be inverted to find $d_{\text{i}}$:

Note that another way to find $d_{\text{i}}$ is to rearrange the equation:

This yields the equation for the image distance as:

Note that there is no inverting here.

The thin lens equations can be used to find the magnification $m$, since both $d_{\text{i}}$ and $d_{\text{o}}$ are known. Entering their values gives

Note that the minus sign causes the magnification to be negative when the image is inverted. Ray tracing and the use of the thin lens equations produce consistent results. The thin lens equations give the most precise results, being limited only by the accuracy of the given information. Ray tracing is limited by the accuracy with which you can draw, but it is highly useful both conceptually and visually.