(III) A bright object is placed on one side of a converging lens of focal length f, and a white screen for viewing the image is on the opposite side. The distance d_T = dᵢ + dₒ between the object and the screen is kept fixed, but the lens can be moved.

(a) Show that if d_T > 4ƒ, there will be two positions where the lens can be placed and a sharp image will be produced on the screen.

(b) If d_T < 4ƒ, show that there will be no lens position where a sharp image is formed.

(c) Determine a formula for the distance between the two lens positions in part (a), and the ratio of the image sizes.

Verified step by step guidance

1. (a) Start by using the lens formula \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(d_o\) is the object distance and \(d_i\) is the image distance. Since \(d_T = d_o + d_i\) is constant, express \(d_i\) in terms of \(d_o\) and \(d_T\): \(d_i = d_T - d_o\). Substitute this into the lens formula to get \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_T - d_o}\).
2. (a) Simplify the equation to find the quadratic equation in terms of \(d_o\): Multiply through by \(d_o(d_T - d_o)\) to eliminate the denominators, resulting in \(d_o(d_T - d_o) = f(d_T - 2d_o)\). Rearrange and simplify to get a quadratic equation: \(d_o^2 - d_Td_o + 2fd_o - fd_T = 0\).
3. (a) Solve the quadratic equation for \(d_o\) using the quadratic formula: \(d_o = \frac{d_T \pm \sqrt{d_T^2 - 8fd_T}}{2}\). For real and distinct solutions (indicating two positions of the lens), the discriminant (inside the square root) must be positive: \(d_T^2 - 8fd_T > 0\). Solve this inequality to find that \(d_T > 4f\) is required.
4. (b) Using the same discriminant condition from part (a), if \(d_T < 4f\), then \(d_T^2 - 8fd_T < 0\), making the discriminant negative. This implies that there are no real solutions to the quadratic equation, meaning no position of the lens will produce a sharp image.
5. (c) To find the distance between the two lens positions, calculate the difference between the two solutions for \(d_o\) from part (a): \(\Delta d_o = \left|\frac{\sqrt{d_T^2 - 8fd_T}}{2}\right|\). To find the ratio of the image sizes, use the magnification formula \(m = -\frac{d_i}{d_o}\) for each position and find the ratio of these magnifications.