Question

A convex lens of focal length \(f\) produces a real image \(x\) times the size of an object, Then what is the distance of the object from the lens? (A) \((\mathrm{x}+1) \mathrm{f}\) (B) \((\mathrm{x}-1) \mathrm{f}\) (C) \([(\mathrm{x}+1) / \mathrm{x}] \mathrm{f}\) (D) \([(x-1) / x] f\)

Short answer

The distance of the object from the lens is \( u = \frac{(x-1)f}{x} \).

Step-by-step solution

Write down the lens formula and the magnification formula.

First, write down the lens formula and magnification formula, which will be used together to determine the object's distance from the lens. Lens formula: \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \) Magnification formula: \( m = -\frac{v}{u} \)

Replace m with x in the magnification formula

Since m = x, we can write the magnification formula using x: \( x = -\frac{v}{u} \)

Solve for v in the magnification formula.

Now, solve for v in the magnification formula: \( v = -xu \)

Substitute v in the lens formula.

Replace v with the expression found in Step 3 in the lens formula: \( \frac{1}{f} = \frac{1}{-xu} - \frac{1}{u} \)

Simplify and solve for u.

Simplify the equation and solve for u: \( \frac{1}{f} = \frac{1-x}{xu} \) Multiplying both sides by f and xu, we get: \( xu = (-x+1)f \) Now, divide both sides by x: \( u = \frac{(-x+1)f}{x} \) Comparing our result with the given options, we find that it matches Option (D). Thus, the distance of the object from the lens is: \( u = \frac{(x-1)f}{x} \).