The correct answer is option 'A': three times its shear modulus.

To understand why, let's first define what Poisson's ratio and elastic modulus are.

Poisson's ratio (ν) is a measure of the lateral deformation that occurs when a material is subjected to axial loading. It is defined as the ratio of the transverse strain (εt) to the axial strain (εa) in a material.

Poisson's ratio (ν) = -εt/εa

On the other hand, the elastic modulus (E) is a measure of the stiffness or rigidity of a material. It represents the ratio of stress (σ) to strain (ε) in a material under elastic deformation.

Elastic modulus (E) = σ/ε

Now, let's solve the problem step by step:

1. Poisson's ratio (ν) is given as 0.5, which means that the transverse strain (εt) is equal to half of the axial strain (εa) in the material.

2. Using the definition of Poisson's ratio, we can rewrite it as:

0.5 = -εt/εa

Rearranging the equation, we get:

εt = -0.5εa

This means that the lateral deformation (εt) is equal to half of the axial deformation (εa) but in the opposite direction.

3. Now, consider a material subjected to shear stress. Shear stress (τ) is defined as the ratio of shear force (F) to the area (A) over which the force is applied.

Shear stress (τ) = F/A

4. Shear strain (γ) is defined as the ratio of the displacement (Δx) to the original height (h) of the material.

Shear strain (γ) = Δx/h

5. The shear modulus (G) is a measure of the rigidity of a material under shear stress. It represents the ratio of shear stress (τ) to shear strain (γ).

Shear modulus (G) = τ/γ

6. Now, consider a material with Poisson's ratio (ν) of 0.5 subjected to shear stress. Due to the lateral deformation, the material will experience a shear strain (γ) and shear stress (τ).

7. From the definition of Poisson's ratio, we know that the lateral deformation (εt) is equal to half of the axial deformation (εa) but in the opposite direction.

8. This means that the lateral deformation (εt) is equal to the shear strain (γ).

9. Therefore, we can write:

Shear modulus (G) = τ/γ = τ/εt

10. From the definition of elastic modulus (E), we know that stress (σ) is equal to shear stress (τ) and strain (ε) is equal to lateral deformation (εt).

11. Therefore, we can write:

Elastic modulus (E) = σ/ε = τ/εt

12. Comparing the equations for shear modulus (G) and elastic modulus (E), we can see that:

Shear modulus (G) = Elastic modulus (E)

13. Since the elastic modulus (E) is equal to the shear modulus (G), option 'C' is incorrect.

14. However, the problem states that the correct answer is

E=2C(1+2u)
given,u=0.5 then E=2C(1+0.5)
=>E=3C that means Young's modulus is three times of shear modulus.