The cross-sections of wires made of the same | vedclass

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Question

(A) The resistance of a wire is given by $R = \rho \frac{l}{A}$,where $\rho$ is the resistivity,$l$ is the length,and $A$ is the cross-sectional area.

Vedclass · Accepted Answer

$R_A = R_B = R_C$

Vedclass · Answer

(A) The resistance of a wire is given by $R = \rho \frac{l}{A}$,where $\rho$ is the resistivity,$l$ is the length,and $A$ is the cross-sectional area.Since all wires are made of the same material ($\rho$ is constant) and have the same length ($l$ is constant),the resistance depends only on the cross-sectional area $A$.For wire $A$: The cross-section is a square frame with outer side $\sqrt{3}a$ and inner side $\sqrt{2}a$. Area $A_A = (\sqrt{3}a)^2 - (\sqrt{2}a)^2 = 3a^2 - 2a^2 = a^2$.For wire $B$: The cross-section is a square frame with outer side $\sqrt{2}a$ and inner side $a$. Area $A_B = (\sqrt{2}a)^2 - (a)^2 = 2a^2 - a^2 = a^2$.For wire $C$: The cross-section is a solid square with side $a$. Area $A_C = a^2$.Since $A_A = A_B = A_C = a^2$,it follows that $R_A = R_B = R_C$.

The cross-sections of wires made of the same material and having the same length are shown in the figure. What can be said about their resistances?

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