The work done in twisting a steel wire of length 25 cm and radius 2mm through 45° will be (η = 8 x 10¹⁰ N / m²)

The correct option is 2.48 J.

To find the work done in twisting a steel wire, we can use the formula for the work done in twisting a cylinder or wire:
$W=\frac{1}{2}C{\theta }^2$
where $C$ is the torsional rigidity (restoring couple per unit twist) of the wire, and $\theta$ is the angle of twist in radians.

The restoring couple per unit twist ($C$) is given by the formula:
$C=\frac{\pi \eta r^4}{2L}$
where:
• $\eta$ is the modulus of rigidity = $8\times {10}^{10}{\text{N/m}}^2$
• $r$ is the radius of the wire = $2\text{mm}=2\times {10}^{-3}\text{m}$
• $L$ is the length of the wire = $25\text{cm}=0.25\text{m}$
• $\theta$ is the angle of twist = ${45}^{\circ }=45\times \frac{\pi }{180}=\frac{\pi }{4}\text{rad}$

First, let's calculate the value of $C$:
$C=\frac{\pi \times (8\times {10}^{10})\times {(2\times {10}^{-3})}^4}{2\times 0.25}$
$C=\frac{\pi \times 8\times {10}^{10}\times 16\times {10}^{-12}}{0.5}$
$C=2\times \pi \times 128\times {10}^{-2}$
$C=2.56\pi \text{N}·\text{m/rad}$

Now, let's calculate the work done ($W$):
$W=\frac{1}{2}\times 2.56\pi \times {\left(\frac{\pi }{4}\right)}^2$
$W=1.28\pi \times \frac{{\pi }^2}{16}$
$W=0.08{\pi }^3$
Using the approximation ${\pi }^3\approx 31.006$:
$W\approx 0.08\times 31.006\approx 2.48\text{J}$

Thus, the work done in twisting the steel wire is approximately 2.48 J.