One mile of copper wire has a resistance of 13 Ω. Using the wire to electrically connect two terminals that are half a mile apart, rank the following situations in order of increasing resistance.
I. Use the full length of wire.
II. Cut the wire in half and use only one half.
III. Cut the wire in half and use both halves, connecting both wire ends at both terminals.
To determine the order of increasing resistance for the different situations, let's analyze each scenario step-by-step:
I. Use the full length of wire:
When using the full length of wire, the resistance is given as 13 Ω for one mile of copper wire. Since the wire is half a mile long in this situation, the resistance would be half of the resistance of one mile. Therefore, the resistance for this scenario is 13 Ω / 2 = 6.5 Ω.
II. Cut the wire in half and use only one half:
When the wire is cut in half and only one half is used, the resistance is determined by the length of wire being utilized. Since only half a mile of wire is used, the resistance would also be halved compared to one mile. Therefore, the resistance for this scenario is 13 Ω / 2 = 6.5 Ω.
III. Cut the wire in half and use both halves, connecting both wire ends at both terminals:
In this case, the resistance is affected by the fact that both wire halves are being used. Connecting both wire ends at both terminals results in the resistance being halved twice. Hence, the resistance for this scenario is 13 Ω / 2 / 2 = 3.25 Ω.
Ranking the situations from increasing resistance:
Comparing the resistance values obtained:
I. Use the full length of wire: 6.5 Ω
II. Cut the wire in half and use only one half: 6.5 Ω
III. Cut the wire in half and use both halves, connecting both wire ends at both terminals: 3.25 Ω
Therefore, the ranking of the situations in order of increasing resistance is: III < II < I.
To rank the situations in order of increasing resistance, we need to consider how the resistance changes when the wire is cut or used in different configurations.
First, let's analyze the resistance for each situation:
I. Use the full length of wire:
Since the full length of the wire is used, the resistance will be equal to the initial resistance of the entire wire, which is given as 13 Ω.
II. Cut the wire in half and use only one half:
When the wire is cut in half and only one half is used, the resistance will decrease. This is because the resistance is directly proportional to the length of the wire. Using only one half of the wire reduces the length to half a mile, which means the resistance will also be halved. Therefore, the resistance in this situation would be 13 Ω / 2 = 6.5 Ω.
III. Cut the wire in half and use both halves, connecting both wire ends at both terminals:
In this situation, the two halves of the wire are used simultaneously, effectively doubling the total length of the wire to one mile again. Since the resistance is directly proportional to the length of the wire, using both halves would result in twice the initial resistance. Therefore, the resistance in this situation would be 2 x 13 Ω = 26 Ω.
Based on the analysis above, the situations can be ranked in order of increasing resistance as follows:
II. Cut the wire in half and use only one half (6.5 Ω)
I. Use the full length of wire (13 Ω)
III. Cut the wire in half and use both halves, connecting both wire ends at both terminals (26 Ω)