91Ó°ÊÓ

Use the formula of power in relation with potential difference and resistance

$P=\frac{V^2}{R}$

Rearrange the equation to find the resistance of the cable and substitute the values

$$\begin{gathered} R=\frac{V^2}{P} \\ R=\frac{{\left(200V\right)}^2}{90W} \\ R=537\mathrm{Î\copyright } \end{gathered}$$

Now, the resistance of the wire depends on the area of the wire, resistivity, and the length of the cable, thus, use the formula in relation with these variables

$$\begin{gathered} R=\frac{\mathrm{ÒÏ}L}{\mathrm{Ï€}{r}^2} \\ r=\sqrt{\frac{\mathrm{ÒÏ}L}{\mathrm{Ï€}R}} \end{gathered}$$

Now, substitute the values

$$\begin{gathered} r=\sqrt{\frac{\left(1.72×{10}^{-8}\mathrm{Î\copyright }·m\right)\left(1500m\right)}{\mathrm{Ï€}\left(537\mathrm{Î\copyright }\right)}} \\ r=0.123 \\ ⇒d=2r \\ ⇒d=\left(2\right)\left(0.123\right) \\ ⇒d=0.246 \end{gathered}$$

Therefore, the diameter of the copper cable should be $0.246mm$so that it produces heat at a rate of $90.0-W$

Use the formula of electric field in relation with potential difference and length of the wire

$$\begin{gathered} E=\frac{V}{L} \\ E=\frac{220V}{1500m} \\ E=0.147V/m \end{gathered}$$

Therefore, the electric field inside the cable under these conditions is 0147 V/m