A rod, of length L at room temperature and uniform area of cross section A, is made of a metal having a coefficient of linear expansion \[^0C\]. It is observed that an external compressive force F, applied on each of its ends, prevents any change in the length of the rod, when its temperature rises by \[\vartriangle TK\]. Young's modulus, Y, for this metal is:
A.)\[\dfrac{F}{{2A\alpha \vartriangle T}}\]
B.)\[\dfrac{F}{{A\alpha (\vartriangle T - 273)}}\]
C.)\[\dfrac{F}{{A\alpha \vartriangle T}}\]
D.)\[\dfrac{{2F}}{{A\alpha \vartriangle T}}\]

Hint: Young's module (also known as the Elastic Modulus or Tensile Modulus), is a measure of the mechanical properties of rigid elastic solids, such as tubes, wires, etc. There are other numbers that give us a sense of a material's elastic properties, such as bulk modulus and shear modulus, but Young's Modulus factor is more widely used. That is because it offers us knowledge on a material's tensile elasticity.

Complete step-by-step answer:

Formula used: y\[{\text{ = }}\dfrac{{{\text{F / A}}}}{{(\vartriangle {\text{ }}\ell /\ell )}}\]

Young’s Modulus y = \[\dfrac{{{\text{Stress}}}}{{{\text{Strain}}}}\]

                                       \[{\text{ = }}\dfrac{{{\text{F / A}}}}{{(\vartriangle {\text{ }}\ell /\ell )}}\]

Thermal strain \[\dfrac{{\vartriangle {\text{ }}\ell }}{\ell } = \alpha \vartriangle T\]

                                       \[{\text{ = }}\dfrac{{{\text{F }}}}{{A(\alpha \vartriangle T)}}\]

Hence, option C is the right answer.

Note: Young's modulus describes the connection between stress and stress. The Young's modulus is named after the British physicist Thomas Young. A solid object deforms when a certain load is applied to it. When an object becomes elastic, the body regains its original shape when the pressure is removed.

Compressive force happens when an object is pressed inwards by brute force, forcing it to compress. During this process the relative positions of the object's atoms and molecules shift. The adjustment can be reversible or lasting depending on the sort of substance the compressive force is getting. Depending on the orientation or location of the material where the compressive force is applied, there can often be various effects.