In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is 10 ± 0.1 cm and the distance of its real image from the lens is 20 ± 0.2 cm. The error in the determination of focal length of the lens is n%. The value of n is .

The correct answer is 1.5.

Step 1: Identify the given values and their errors
Distance of the object from the convex lens, $u=-10\text{cm}$ (by sign convention), so $\left|u\right|=10\text{cm}$ and error $\Delta u=0.1\text{cm}$.
Distance of the real image from the lens, $v=20\text{cm}$ and error $\Delta v=0.2\text{cm}$.

Step 2: Calculate the focal length (f) of the convex lens
Using the lens formula:

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

Substituting $u=-10\text{cm}$ and $v=20\text{cm}$:

$\frac{1}{f}=\frac{1}{20}-\left(-\frac{1}{10}\right)=\frac{1}{20}+\frac{1}{10}=\frac{1+2}{20}=\frac{3}{20}{\text{cm}}^{-1}$

Therefore, focal length:

$f=\frac{20}{3}\text{cm}$

Step 3: Derive the formula for absolute error in focal length (Δf)
Differentiating the lens formula $\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$:

$-\frac{df}{f^2}=-\frac{dv}{v^2}+\frac{du}{u^2}$

For maximum error combination, we take positive signs for all individual errors:

$\frac{\Delta f}{f^2}=\frac{\Delta v}{v^2}+\frac{\Delta u}{u^2}$

Multiplying both sides by $f$, the fractional error in focal length is given by:

$\frac{\Delta f}{f}=f\left(\frac{\Delta v}{v^2}+\frac{\Delta u}{u^2}\right)$

Step 4: Calculate the percentage error in focal length (n%)
Percentage error $n\%=\frac{\Delta f}{f}\times 100\%$

Substitute the given values into the fractional error formula:

$\frac{\Delta f}{f}=\left(\frac{20}{3}\right)\times \left(\frac{0.2}{{20}^2}+\frac{0.1}{{10}^2}\right)$

$\frac{\Delta f}{f}=\left(\frac{20}{3}\right)\times \left(\frac{0.2}{400}+\frac{0.1}{100}\right)$

$\frac{\Delta f}{f}=\left(\frac{20}{3}\right)\times \left(0.0005+0.001\right)=\left(\frac{20}{3}\right)\times 0.0015$

$\frac{\Delta f}{f}=\frac{0.03}{3}=0.01$

Now, calculate percentage error:

$n\%=0.01\times 100\%=1\%$

Hence, the value of $n$ is 1.