The graph shows how the magnification m produced by a thin lens varies with image distance v. What is the focal length of the lens used?

To find the focal length of the lens using the graph of magnification $m$ versus image distance $v$, we will go through the following steps.

1. ### Break Down the Problem

1. Identify the relationship between magnification $m$, image distance $v$, and object distance $u$ using the lens formula.
2. Recall the thin lens equation and magnification formulas involved.

2. ### Relevant Concepts

1. The thin lens formula is given by: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ where $f$ is the focal length of the lens, $v$ is the image distance, and $u$ is the object distance.

2. The magnification $m$ can also be expressed as: $m = -\frac{v}{u}$

3. ### Analysis and Detail

1. Rearrange the magnification formula to find object distance $u$: $u = -\frac{v}{m}$

2. Substitute this expression for $u$ into the thin lens formula: $\frac{1}{f} = \frac{1}{v} + \frac{1}{-\frac{v}{m}}$ Simplifying this gives: $\frac{1}{f} = \frac{1}{v} - \frac{m}{v} = \frac{1 - m}{v}$

3. This leads to: $f = \frac{v}{1 - m}$

4. ### Verify and Summarize

1. Using the values of $v$ and $m$ from the graph, substitute them into the calculated equation for $f$ to determine the focal length.
2. Check for multiple points on the graph for consistency.

Final Answer

• Once you have substituted the appropriate values for $v$ and $m$, solve for $f$ to get the focal length of the lens used. Make sure to check at least two points from the graph to ensure accuracy.