A steel rod of length 1 m and cross-sectional area $$10^{-4}$$ m$$^2$$ is heated from 0$$^\circ$$C to 200$$^\circ$$C without being allowed to extend or bend. The compressive tension produced in the rod is _______ $$\times 10^4$$ N. (Given Young's modulus of steel $$= 2 \times 10^{11}$$ N m$$^{-2}$$, coefficient of linear expansion $$= 10^{-5}$$ K$$^{-1}$$)

We have a rod of length $$L = 1$$ m, cross-sectional area $$A = 10^{-4}$$ m$$^2$$, with a temperature change $$\Delta T = 200°C$$, Young's modulus $$Y = 2 \times 10^{11}$$ N/m$$^2$$, and coefficient of linear expansion $$\alpha = 10^{-5}$$ K$$^{-1}$$.

When a rod is heated but not allowed to expand, thermal stress develops. The thermal strain is

$$\text{Strain} = \alpha \Delta T = 10^{-5} \times 200 = 2 \times 10^{-3}$$

Now the compressive stress is (since stress equals Young's modulus times strain)

$$\text{Stress} = Y \times \text{Strain} = 2 \times 10^{11} \times 2 \times 10^{-3} = 4 \times 10^{8} \text{ N/m}^2$$

So the compressive force is

$$F = \text{Stress} \times A = 4 \times 10^{8} \times 10^{-4} = 4 \times 10^{4} \text{ N}$$

Hence, the correct answer is $$4 \times 10^4$$ N.