A steel rod with $$y = 2.0 \times 10^{11}$$ N m$$^{-2}$$ and $$\alpha = 10^{-5}$$ °C$$^{-1}$$ of length 4 m and area of cross-section 10 cm$$^2$$ is heated from 0°C to 400°C without being allowed to extend. The tension produced in the rod is $$x \times 10^5$$ N where the value of $$x$$ is _________.

We begin with the idea that, when a rod is heated but its length is not allowed to change, the thermal tendency to expand is opposed by an internal restoring force. The result is a thermal stress. For a uniform rod that is perfectly constrained, the magnitude of this stress is given by the well-known relation

$$\sigma = Y \, \alpha \, \Delta T,$$

where $$\sigma$$ is the stress, $$Y$$ is Young’s modulus, $$\alpha$$ is the coefficient of linear expansion, and $$\Delta T$$ is the change in temperature.

We have the numerical data: $$Y = 2.0 \times 10^{11}\, \text{N m}^{-2}, \quad \alpha = 10^{-5}\, ^\circ\text{C}^{-1}, \quad \Delta T = 400^\circ\text{C}.$$ Substituting these values,

$$\sigma = \left(2.0 \times 10^{11}\right)\!\left(10^{-5}\right)\!\left(400\right).$$

First multiply $$10^{-5}$$ and $$400$$:

$$10^{-5} \times 400 = 4 \times 10^{-3}.$$

Next multiply this result by $$2.0 \times 10^{11}$$:

$$\sigma = 2.0 \times 10^{11} \times 4 \times 10^{-3} = 8 \times 10^{8}\, \text{N m}^{-2}.$$

So the thermal stress inside the rod is $$\sigma = 8 \times 10^{8}\, \text{N m}^{-2}.$$

The tension (force) in the rod is obtained from the definition $$\sigma = F/A,$$ where $$A$$ is the cross-sectional area. Rearranging, $$F = \sigma A.$$ The given area is $$10 \text{ cm}^2.$$ Converting to square metres,

$$10 \text{ cm}^2 = 10 \times 10^{-4} \text{ m}^2 = 10^{-3} \text{ m}^2.$$

Now substitute $$\sigma = 8 \times 10^{8}\, \text{N m}^{-2}$$ and $$A = 10^{-3} \text{ m}^2$$ into $$F = \sigma A$$:

$$F = \left(8 \times 10^{8}\right) \left(10^{-3}\right) = 8 \times 10^{5}\, \text{N}.$$

The question states that this force can be written as $$x \times 10^{5}\, \text{N}.$$ Comparing, we see $$x = 8.$$

Hence, the correct answer is Option 8.