Resistance of Wire
When a current flows through a piece of wire, the current experiences resistance. If the wire is twice as long, the resistance is twice as much. This seems reasonable, since the charge has to travel twice the distance.
We can understand this better with Ohm's law: R = V/I. Doubling the length is like having two pieces of the wire joined end to end. For the same current I, the potential difference across each piece is V. To move charge through one piece after another, the work done is twice as much. So the potential difference is 2V. Therefore V/I = R becomes 2V/I = 2R.
What if the area of the cross-section is twice as big? In this case, the resistance is halved.
Doubling the cross-section is as if two pieces of wire are attached side by side, rather than end to end. Both pieces are under the same potential difference V. If each piece takes a current I, the resistance of each piece is still R = V/I.
However, two pieces carry a total current of 2I. So the combined effect is an effective resistance of V / (2I) = (V/I) / 2 = R/2. Therefore resistance is halved.
We could imagine that in this case, increasing the cross-sectional area gives the current more room to move through, so it feels less resistance.
Summarising:
The resistance of a uniform wire is directly proportional to its length, and inversely proportional to its cross-sectional area.
Exercise. A piece of metal wire is 1 m long. The area of its cross-section is 1 cm2. Its resistance is 2 Ohms. The wire is cut into two pieces, each of length 0.5 m. What is the resistance of each piece?
Answer. Resistance is directly proportional to the length. If the length is halved, the resistance is halved. Half of 2 Ohms is 1 Ohm. So the resistance of each piece is 1 Ohm.
Exercise. In the above exercise, a piece of wire has been cut into two pieces of the same length. These two pieces are placed side by side, touching each other lengthwise. What is the resistance of this combination?
Answer. The length of each piece is 0.5 m and the cross-sectional area of 1 cm2. The resistance is 1 Ohm. The two pieces side by side has a total cross-sectional area of 2 cm2. Resistance is inversely proportional to the area. Since the area is doubled, the resistance is halved. So 1 Ohm becomes 0.5 Ohm. The resistance of the two pieces side by side is 0.5 Ohm.