The distance between an object and its real image in a convex lens is 40cm. If the magnification of the image is 3, calculate the focal length of the lens

Analysis

We are given:

• The total distance between the object ($u$) and the real image ($v$) is $u + v = 40\text{ cm}$.
• The magnification ($m$) is $3$.
• We need to find the focal length ($f$).

Concept

1. Magnification formula: $m = \frac{v}{u}$. Since $m = 3$, we have $\frac{v}{u} = 3$, which means $v = 3u$.
2. Lens formula: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$.

Step-by-Step Solution

Step 1: Find the values of $u$ and $v$. We know $u + v = 40$. Substitute $v = 3u$ into the equation: $u + 3u = 40$ $4u = 40$ $u = 10\text{ cm}$

Now, find $v$: $v = 3u = 3(10) = 30\text{ cm}$

Step 2: Calculate the focal length ($f$) using the lens formula. $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ $\frac{1}{f} = \frac{1}{10} + \frac{1}{30}$

To add these fractions, find the common denominator (which is 30): $\frac{1}{f} = \frac{3 + 1}{30}$ $\frac{1}{f} = \frac{4}{30}$

Step 3: Solve for $f$. $f = \frac{30}{4}$ $f = 7.5\text{ cm}$

Conclusion

The focal length of the lens is 7.5 cm.

Exam Tip

When dealing with real images in convex lenses, remember that magnification $m$ is always positive. If the question had mentioned a virtual image, the magnification would be treated as negative ($m = -3$), which would change your calculation for $v$. Always check if the image is real or virtual before starting your calculation!