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 _______.

 To determine the error in the focal length of a thin convex lens, we'll use the lens formula and error propagation techniques.

Given:

• Object distance, $u=10\pm 0.1$ cm

• Image distance, $v=20\pm 0.2$ cm

Lens Formula:

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

where $f$ is the focal length.

Step 1: Calculate the focal length $f$

$$\frac{1}{f}=\frac{1}{20}+\frac{1}{10}=\frac{1}{20}+\frac{2}{20}=\frac{3}{20}$$$$f=\frac{20}{3}\approx 6.6667\text{ cm}$$

Step 2: Determine the relative errors

• Relative error in $u$: $\frac{\mathrm{\Delta }u}{u}=\frac{0.1}{10}=0.01$ (1%)

• Relative error in $v$: $\frac{\mathrm{\Delta }v}{v}=\frac{0.2}{20}=0.01$ (1%)

Step 3: Apply error propagation
The relative error in $f$ can be approximated by:

$$\frac{\mathrm{\Delta }f}{f}\approx \sqrt{{\left(\frac{\mathrm{\Delta }u}{u}\right)}^2+{\left(\frac{\mathrm{\Delta }v}{v}\right)}^2}$$$$\frac{\mathrm{\Delta }f}{f}\approx \sqrt{(0.01{)}^2+(0.01{)}^2}=\sqrt{0.0001+0.0001}=\sqrt{0.0002}\approx 0.01414$$$$\frac{\mathrm{\Delta }f}{f}\approx 1.414\mathrm{\%}$$

Step 4: Calculate the absolute error in $f$

$$\mathrm{\Delta }f=f\times \frac{\mathrm{\Delta }f}{f}\approx 6.6667\times 0.01414\approx 0.0943\text{ cm}$$

Step 5: Determine the percentage error $n\mathrm{\%}$

$$n=\frac{\mathrm{\Delta }f}{f}\times 100\approx 1.414\mathrm{\%}$$

Rounding to the nearest whole number, $n\approx 1\mathrm{\%}$.

However, considering the significant figures and the nature of the problem, the closest value to $1.414\mathrm{\%}$ is $1\mathrm{\%}$.

Final Answer:

$$\boxed{{\displaystyle 1}}$$