Question

If the object and image distances due to a convex lens are x each, then its focal length is

• A. 2x
• B. \(\frac{x}{2}\)
• C. \(\frac{2x}{3}\)
• D. 4x

Answer 1

The Correct Option is B

To solve for the focal length of a convex lens when the object and image distances are both equal to \(x\), we use the lens formula:

\[\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\]

where:

\(f\) = focal length of the lens

\(v\) = image distance

\(u\) = object distance

Given that \(v = x\) and \(u = x\), we substitute these values into the lens formula:

\[\frac{1}{f} = \frac{1}{x} + \frac{1}{x}\]

\[\frac{1}{f} = \frac{2}{x}\]

Taking the reciprocal to solve for \(f\):

\[f = \frac{x}{2}\]

Thus, the focal length of the convex lens is \(\frac{x}{2}\).

Answer 2

Let the object distance be u = –x (negative as it's on the left of the lens), and the image distance be v = +x.

Using the lens formula: 1/f = 1/v – 1/u

Substituting the values: 1/f = 1/x – (–1/x) = 1/x + 1/x = 2/x

So, the focal length f = x/2.

Final Answer: x/2