A cable is replaced by another one of the same length and material but of twice the diameter. The maximum load that the new wire can support without exceeding the elastic limit, as compared to the load that the original wire could support, is

The correct option is Four times.

To understand why the maximum load increases by a factor of four, let us analyze the relation between stress, load, and the cross-sectional area of a cable.

The elastic limit of a material is defined in terms of the maximum stress (force per unit area) the material can withstand without undergoing permanent deformation. This maximum stress, known as the elastic limit stress (${\sigma }_{\text{max}}$), is a characteristic property of the material itself. Since the replacement cable is made of the same material as the original cable, both cables have the same elastic limit stress:

${\sigma }_{\text{max, new}}={\sigma }_{\text{max, original}}$

The stress $\sigma$ in a cable supporting a load $F$ is given by the formula:
$\sigma =\frac{F}{A}$
where $A$ is the cross-sectional area of the cable.

Therefore, the maximum load $F_{\text{max}}$ that a cable can support without exceeding its elastic limit is:
$F_{\text{max}}={\sigma }_{\text{max}}\cdot A$

For a cable with a circular cross-section of diameter $d$, the cross-sectional area is:
$A=\frac{\pi }{4}{d}^2$

Substituting this area into the maximum load equation gives:
$F_{\text{max}}={\sigma }_{\text{max}}\cdot \frac{\pi d^2}{4}$

This shows that the maximum load is directly proportional to the square of the diameter ($F_{\text{max}}\propto d^2$).

Let the diameter of the original cable be $d_1$ and the diameter of the new cable be $d_2=2d_1$. The ratio of the maximum load supported by the new cable ($F_{\text{max, new}}$) to that of the original cable ($F_{\text{max, original}}$) is:
$\frac{F_{\text{max, new}}}{F_{\text{max, original}}}=\frac{{\left(d_2\right)}^2}{{\left(d_1\right)}^2}=\frac{{\left(2d_1\right)}^2}{{d_1}^2}=\frac{4{d_1}^2}{{d_1}^2}=4$

Thus, the maximum load that the new wire can support without exceeding its elastic limit is four times the load that the original wire could support.