Focal Length of a Converging Lens: Thin Lens Approximation and Systematic Errors
A student determines the focal length of a converging lens using two methods. Method A: They focus a distant object (a building 200 m away) onto a screen and measure the distance from the lens holder to the screen as 15.2 cm. Method B: They place an object at 30.0 cm from the lens holder and find the sharp image on a screen at 30.5 cm from the lens holder. They also plot a graph of 1/v against 1/u for several object distances and obtain a straight line with a y-intercept of 0.065 cm⁻¹ and a gradient of -1.02. The lens has a thickness of 0.8 cm, and all distances were measured from the centre of the lens holder, not the optical centre. The screen has a thickness of 0.2 cm. The student uses the thin lens equation 1/f = 1/u + 1/v.
1. Calculate the focal length using Method A and Method B, and compare with the value from the graph.
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Method A: For a distant object, u ≈ ∞, so 1/u ≈ 0. The thin lens equation gives 1/f ≈ 1/v, so f ≈ v = 15.2 cm.
Method B: Using u = 30.0 cm and v = 30.5 cm, 1/f = 1/u + 1/v = 1/30.0 + 1/30.5 = 0.03333 + 0.03279 = 0.06612 cm⁻¹, so f = 1/0.06612 = 15.12 cm.
Graph method: The graph of 1/v against 1/u has the form 1/v = −(1/u) + 1/f, so the y-intercept is 1/f. The y-intercept is 0.065 cm⁻¹, giving f = 1/0.065 = 15.38 cm. The gradient is −1.02, which is close to the theoretical −1 for a thin lens, but not exactly −1, indicating some systematic error.
Comparison: Method A gives 15.2 cm, Method B gives 15.1 cm, and the graph gives 15.4 cm. The values are close (within about 0.3 cm), but the graph value is slightly higher.
2. Explain why the values might differ, considering the lens thickness and the screen thickness.
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The lens is not thin (thickness 0.8 cm), so the thin lens equation is only an approximation. For a thick lens, the focal length is measured from the principal planes, not from the lens centre or holder. The distances measured from the lens holder may not correspond to the true object and image distances from the principal planes. This introduces a systematic offset.
The screen thickness (0.2 cm) matters if the image is formed on the front surface of the screen but the distance is measured to the back surface. In that case, the measured image distance is 0.2 cm too large, which would make the calculated focal length slightly too large. However, the effect is small (0.2 cm in 30 cm is about 0.7%).
The gradient of −1.02 instead of −1 suggests a small systematic error in the measurement of u or v, possibly due to the lens thickness or the finite size of the object. The graph method uses many data points, so random errors are reduced, but systematic errors remain. The y-intercept is the most reliable estimate of 1/f because it is less sensitive to errors in individual points, but it still inherits any systematic offset. The differences between the methods (about 0.3 cm) are consistent with these small systematic effects.
3. Suggest a correction for the lens thickness and estimate its effect on the focal length determined from Method B.
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If the lens holder centre is at the front surface of the lens, the optical centre is half the thickness (0.4 cm) behind the front surface. For the object side, the true object distance from the optical centre is u_true = u_measured + 0.4 cm = 30.4 cm. For the image side, the true image distance from the optical centre is v_true = v_measured − 0.4 cm = 30.1 cm. Using these corrected values in the thin lens equation: 1/f = 1/30.4 + 1/30.1 = 0.03289 + 0.03322 = 0.06611 cm⁻¹, so f = 15.13 cm. This is almost identical to the uncorrected value of 15.12 cm, a change of only 0.01 cm. Therefore, the lens thickness correction has a negligible effect on the focal length in this case, because the shifts in u and v partially cancel. A more sophisticated treatment using thick lens formulae would be needed if higher precision were required, but for this experiment the thin lens approximation is adequate.