A wire made from a particular metal has a length of 2.0 m, a cross-sectional area of 0.50 mm^2, and a resistance of 4.0 at constant temperature. (a) A second wire is made from the same metal, with the same cross-sectional area but a length of 5.0 m. Calculate its resistance. [2]…

Answer: ((a)) 10 ((b)) 8.0. Resistance is inversely proportional to cross-sectional area, so halving the area doubles the resistance. ((c)) A thick cable has a larger cross-sectional area and therefore a lower resistance, so less electrical energy is transferred to heat for a given current.. Part (a): Effect of increasing the length For a wire of a given material and cross-sectional area, resistance is directly proportional to length: R L Going from L1 = 2.0 m to L2 = 5.0 m, the resistance scales by the same factor as the leng…

IGCSE Physics 0625: How Wire Length and Cross-Sectional Area Affect Resistance

A wire made from a particular metal has a length of $2.0\text{ m}$ , a cross-sectional area of $0.50\text{ mm}^2$ , and a resistance of $4.0\,\Omega$ at constant temperature.

(a) A second wire is made from the same metal, with the same cross-sectional area but a length of $5.0\text{ m}$ . Calculate its resistance. [2]

(b) A third wire, made from the same metal and with a length of $2.0\text{ m}$ , has half the cross-sectional area of the original wire. State the resistance of this third wire, and explain how you found your answer. [2]

(c) The main cables used to carry large currents from a power station are made thick rather than thin. Explain, in terms of resistance, why this reduces the energy wasted by the cables. [2]

Part (a): Effect of increasing the length

For a wire of a given material and cross-sectional area, resistance is directly proportional to length:

$R \propto L$

Going from $L_1 = 2.0\text{ m}$ to $L_2 = 5.0\text{ m}$ , the resistance scales by the same factor as the length:

$R_2 = R_1 \times \frac{L_2}{L_1} = 4.0\,\Omega \times \frac{5.0}{2.0}$

$R_2 = \boxed{10\,\Omega}$

Part (b): Effect of halving the cross-sectional area

For a wire of a given material and length, resistance is inversely proportional to cross-sectional area:

$R \propto \frac{1}{A}$

Halving the cross-sectional area means the resistance must double, since a thinner wire offers a narrower path for the charge carriers and so opposes the current more:

$R_3 = R_1 \times \frac{A_1}{\tfrac{1}{2}A_1} = R_1 \times 2 = 4.0\,\Omega \times 2$

$R_3 = \boxed{8.0\,\Omega}$

Part (c): Why power cables are made thick

A thicker cable has a larger cross-sectional area. From the proportionality above, a larger cross-sectional area gives a lower resistance (for the same length and material).

A lower resistance means less electrical energy is transferred to heat in the cable for a given current, since the energy wasted as heat increases with the cable’s resistance. Making the cables thick (low resistance) therefore reduces the energy wasted during transmission, so more of the electrical energy generated at the power station reaches its destination usefully.

Final answers

(a) Resistance of the longer wire $=$ $10\,\Omega$
(b) Resistance of the thinner wire $=$ $8.0\,\Omega$ (resistance doubles when cross-sectional area is halved)
(c) Thick cables have a larger cross-sectional area, giving lower resistance and so less energy wasted as heat