Question

Two copper wires of different lengths are used in a circuit. Which wire has more resistance?

Answer 1

Recall that a longer wire length implies a larger number of ionic vibrations that the electrons need to get through. Use this with the definition of electrical resistance, which is nothing but the obstruction imparted to the flow of current, to arrive at how the length of the wire changes their resistance to current/electron flow. Note that since both the wires are made of copper, they possess the same resistivity.

Complete Step-by-Step Solution:
Let us begin by understanding the concept of electrical resistance.

Electrical resistance, or impedance is the opposition offered by the material of the wire to the flow of electricity through the circuit. Thus, the resistance of an object depends largely on the material that it is made of, since objects made of insulators tend to have a very high resistance and those made of conductors tend to have a low resistance.

The cause of such resistance is due to the collision between electrons flowing through the conductor as current and the fixed ions constituting the structure of the conductor.

Consequently, the movement of the electrons is hindered by the atomic lattice vibrations which causes a part of the electrical energy of the electric current to be lost as resistance. So, if the temperature is increased, this increases the lattice vibrations which increases the likelihood of collisions between electrons and atoms leading to more resistance to the flow of current.

This leads us to the definition of resistivity, which is nothing but a measure of resistance specific to a material. Therefore, different materials have different resistivity.
In addition to the material constituting the object, the resistance also depends on the size of the object and the dimensions of the object.

We are given that two copper wires (conductors) of different lengths are used in a circuit. This means that we can eliminate the material dependence of resistance since they’re both made of the same material (copper), which implies they have the same resistivity.

We know that electrical resistance depends on the dimensions of the object since it defines the probable number of collisions the electrons can undergo with the vibrating ions. A longer wire implies that the electrons have to travel a longer length, which means that they have to pass through more atomic lattice vibrations through which they are subjected to relatively more collisions. The increased number of collisions naturally leads to an increase in the resistance of the wire (which may sometimes increase the temperature of the wire, leading to even more frequent lattice vibrations and subsequent collisions, all adding up to an increased resistance of the wire nonetheless).

Thus, we can conclude that resistance is directly proportional to the length of the conductor. $R\propto l$, which means that the longer wire has more resistance.

Note:
Do not get confused between resistance and resistivity.

Resistance is a measure of the opposition that an object offers to a flow of current, whereas resistivity is a characteristic property of the material making up the object which resists the flow of current through it.

Resistance of a wire is dependent on the length, cross-sectional area and temperature of the wire, whereas its resistivity is dependent only on the material and temperature of the wire. Thus, the resistivity of a wire remains the same with a change in the length or area of the wire but changes with variations in temperature.

The two are related as: $R=\rho \dfrac{l}{A}$, where R is the resistance, $\rho$ is the resistivity, l is the length of the conductor and A is its area of cross-section.