Two wires `A and B` are of equal lengths, different cross-sectional areas and made of the same metal.
(a) (i) Name the property which is same for both the wires.
(ii) Name the property which is the different for both the wires
(b) I fthe resistance of wire `A` is four times the resistance of wire `B`, calculate
(i) the ratio of the cross-sectional area of the wires.
(ii) the ratio of the radii of the wires.

Explanation

(a) (i) Resistivity. This is due to the reason that the resistivity is the property of the material of which the wire is made. As both the wires are made of the same metal, their resistivity is the same.
(ii) Resistance. As both the wires are of different cross-sectional area, their resistances are different `(r prop 1//A)`.
(b) (i) For wire A, `R_1 = (rho l)/(A_1)` and for wire B, `R_1 = (rho l)/(A_2)` Thus, `(R_2)/(R_1) = (A_1)/(A_2)`
Since `R_1 = 4 R_2, (R_2)/(R_1) = (1)/(4)` Thus, `(A_1)/(A_2)=(1)/(4)`
(ii) As `A_1 = pi r_1^2` and `A_2 = pi r_2^2, (A_1)/(A_2) = (pi r_1^2)/(pi r_2^2) = ((r_1)/(r_2))^2`
As `(A_1)/(A_2) = (1)/(4), ((r_1)/(r_2))^2 = (1)/(4)` or `(r_1)/(r_2) = (1)/(2)`.