Question

Find the focal length of the given convex lens by plotting graph between u and v.

Hint:

To ensure the accuracy of the u-v graph, take readings for object positions both between f and 2f, and beyond 2f. Removing parallax accurately is the key to obtaining good data points for the graph. The tip of the needles and the optical center of the lens should be at the same height.

Answer 1

Step 1: Understanding the Concept: 
For a convex lens forming a real image, the object distance (u), image distance (v), and focal length (f) are related by the lens formula. A graph plotted between u and v is a rectangular hyperbola. From this graph, the focal length can be determined because, for a convex lens, when the object is placed at a distance of 2f from the optical center, a real image of the same size is formed at a distance of 2f on the other side. Therefore, the point on the graph where u = v corresponds to u = v = 2f. 
Step 2: Key Formula and Apparatus: 
Apparatus Required: 
An optical bench, a convex lens, a lens holder, two optical needles (one for the object, one for the image), and a meter scale. 
Key Formula: 
Lens formula: \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \) 
From the u-v graph, at the point of intersection P with the line u=v, we have: \[ u_P = v_P = 2f \] Therefore, \( f = \frac{u_P}{2} = \frac{v_P}{2} \). 
Step 3: Detailed Procedure: 
1. Rough Focal Length: Find the approximate focal length of the lens by focusing the image of a distant object (like a window) on a screen. 
2. Setup: Mount the lens on the holder on the optical bench. Place the object needle at a distance greater than 2f from the lens. 
3. Data Collection: - Adjust the position of the image needle on the other side of the lens until parallax between the tip of the image needle and the real, inverted image of the object needle is removed. 
- Record the positions of the object needle, lens, and image needle. 
- Calculate the object distance u (distance between object needle and lens) and image distance v (distance between image needle and lens). 
- Repeat the process for 5-6 different values of u, moving the object needle away from the lens. 
Step 4: Graph and Calculation: 
1. Plotting: Plot a graph with u on the X-axis and v on the Y-axis. The graph will be a curve (a rectangular hyperbola). 
2. Finding 2f: Draw a line that passes through the origin at an angle of 45° to the axes (the line u = v). This line will intersect the u-v curve at a point P. 
3. Reading Coordinates: Read the coordinates of the point P from the graph. Let them be \((u_P, v_P)\). At this point, \(u_P = v_P\). 
4. Calculating Focal Length: Calculate the focal length using the relation: \[ f = \frac{u_P}{2} \quad \text{or} \quad f = \frac{v_P}{2} \] The average of these gives the focal length of the lens.