The focal length f of a converging lens can be found by placing an object of known size at various locations in front of the lens and measuring the resulting real-image distances dᵢ and their associated magnifications m (minus sign indicates that image is inverted). The data taken in such an experiment are given here:


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(a) Show algebraically that a graph of m vs. dᵢ should produce a straight line. What are the theoretically expected values for the slope and the y-intercept of this line? [Hint: dₒ is not constant.] (b) Using the data above, graph m vs. dᵢ and show that a straight line does indeed result. Use the slope of this line to determine the focal length of the lens. Does the y-intercept of your plot have the expected value?

Verified step by step guidance

1. Understand the relationship between object distance (dₒ), image distance (dᵢ), and magnification (m). The lens formula is \( \frac{1}{f} = \frac{1}{dₒ} + \frac{1}{dᵢ} \) and the magnification formula is \( m = -\frac{dᵢ}{dₒ} \).
2. Rearrange the magnification formula to express dₒ in terms of dᵢ and m: \( dₒ = -\frac{dᵢ}{m} \). Substitute this expression for dₒ into the lens formula to get a new equation involving f, dᵢ, and m.
3. Simplify the equation from step 2 to isolate f on one side. This will help in understanding how f relates to dᵢ and m, and how changes in dᵢ and m affect f.
4. Plot the data points of m versus dᵢ on a graph. The relationship derived in step 3 will help you understand the expected shape of the graph (straight line) and how to interpret the slope and y-intercept in terms of the lens's focal length.
5. Use the slope from the linear graph to calculate the focal length of the lens. Compare the theoretical y-intercept with the experimental y-intercept to check for consistency with the expected values.

Key Concepts

Lens Formula

The lens formula relates the object distance (dₒ), image distance (dᵢ), and focal length (f) of a lens through the equation 1/f = 1/dₒ + 1/dᵢ. This formula is fundamental in optics, allowing us to understand how light converges through a lens and how the position of the object affects the image formed.

Magnification

Magnification (m) is defined as the ratio of the height of the image (hᵢ) to the height of the object (hₒ), and it can also be expressed as m = -dᵢ/dₒ. This negative sign indicates that the image is inverted. Understanding magnification is crucial for analyzing how the size of the image changes with respect to the object as it moves relative to the lens.

Graphing Relationships

In this experiment, plotting magnification (m) against image distance (dᵢ) should yield a linear relationship, as derived from the lens formula. The slope of this line can be related to the focal length of the lens, while the y-intercept provides additional insights into the system's behavior. Recognizing how to interpret graphs is essential for analyzing experimental data in physics.