Question

Write in the form of formula a relationship between resistance and resistivity of an ohmic conductor. A wire of length \[\text{l}\] and resistance \[\text{R}\] is straightened so that it’s length is doubled and the area of the cross section is halved. How will it’s-
(A). Resistance change
(B). Resistivity change

Answer 1

Hint: The resistance of a conductor is directly proportional to length and inversely proportional to area of cross section. Resistivity is the constant of proportionality in the above relationship. Substituting new values of length and area in the equation will give change in resistance.

Formulas used:
\[\text{R=}\dfrac{\text{ }\!\!\rho\!\!\text{ l}}{\text{A}}\]

Complete step by step solution:
Resistance is the property of a material by virtue of which it opposes the flow of electricity through it. It’s SI unit is ohm \[(\Omega )\].
For a spherical segment of a conductor, with length \[\text{l}\] and area of cross section \[\text{A}\] and resistance \[\text{R}\] the relationship between \[\text{R,A and l}\] is-
\[\text{R}\propto \dfrac{\text{l}}{\text{A}}\]
Removing the sign of proportionality, we get,
\[\text{R=}\dfrac{\text{ }\!\!\rho\!\!\text{ l}}{\text{A}}\] --------- (1)
\[\text{ }\!\!\rho\!\!\text{ }\] is the constant of proportionality, it is also called resistivity. It depends on the material of the conductor and on temperature.
Resistivity is defined as the resistance of a conductor whose length is \[\text{1m}\] and area of cross section is\[\text{1}{{\text{m}}^{\text{2}}}\]
When the properties of a conductor are changed as-
\[\text{l }\!\!'\!\!\text{ = 2l}\] ----------- (2)
\[\text{A }\!\!'\!\!\text{ = }\dfrac{\text{A}}{2}\] ------------ (3)
The resistance of the conductor will change as-
From eq (1), eq (2) and eq (3), we get,
\[\text{R }\!\!'\!\!\text{ = }\dfrac{\text{ }\!\!\rho\!\!\text{ (2l)}}{\dfrac{A}{2}}\]
\[\text{R }\!\!'\!\!\text{ = }\dfrac{\text{4 }\!\!\rho\!\!\text{ l}}{\text{A}}\] - (4)
Comparing eq (1) and eq (4), we get,
\[\text{R }\!\!'\!\!\text{ = 4R}\]
The new resistance is 4 times greater than the original resistance.
The resistivity will not change with change in the length and area of the cross section of the conductor because it is a constant quantity and does not depend on the length and area. It depends on the type of material of the conductor and temperature.
Therefore the relationship between resistance and resistivity is given by \[\text{R=}\dfrac{\text{ }\!\!\rho\!\!\text{ l}}{\text{A}}\].
When length is doubled and area is halved, the resistance of the conductor becomes \[\text{4R}\] but resistivity does not change.

Note:
Resistivity is the characteristic property of a material and does not depend on it’s dimensions. It’s SI unit is ohm-metre\[\text{( }\!\!\Omega\!\!\text{ -m)}\]. The resistance of a material decreases with increase in temperature, so resistivity also decreases. Resistance also depends on current, \[\text{I}\] and voltage,\[\text{V}\] as- \[\text{R = }\dfrac{\text{V}}{\text{I}}\].