Which has a greater resistivity?
1. A copper rod that is shaped like a rectangular prism with dimensions of 1cm x 1cm x 1000m
2. A copper rod that is shaped like a solid cylinder with diameter 1cm
and length 1000m
I think it should be the rectangular prism because it has a greater area. Is this right?
Or do they both have the same resistivity?
The skinnier the wire is the greater its resistance. The fatter it is, the less the resistance.
What about for resistivity though?
the resistivity of the copper is constant
To determine which of the two copper rods has a greater resistivity, we need to understand the formula for resistivity and how it relates to the shape and dimensions of a conductor.
The resistivity (ρ) of a material is a constant that quantifies how strongly it resists the flow of electric current. It is dependent on the material's intrinsic properties and is given by the formula:
ρ = R × (A / L)
Where:
- ρ is the resistivity of the material,
- R is the resistance of the conductor made from the material,
- A is the cross-sectional area of the conductor,
- L is the length of the conductor.
From the given information, we have two copper rods with different shapes and dimensions:
1. Copper rod with rectangular prism shape: dimensions 1cm x 1cm x 1000m.
2. Copper rod with a solid cylinder shape: diameter 1cm and length 1000m.
Using the formula for resistivity, let's compare the resistivity values of both rods.
For the rectangular prism:
- Cross-sectional area (A) = length (1 cm) * width (1 cm) = 1 cm^2
- Length (L) = 1000 m
For the solid cylinder:
- Cross-sectional area (A) = pi * (radius)^2 = pi * (0.5 cm)^2 = 0.785 cm^2
(Note: The radius is half the diameter.)
- Length (L) = 1000 m
Now, to find the resistivity ratio, we need to compare the values of ρ for both rods.
For the rectangular prism, let's substitute the values:
ρ1 = R * (1 cm^2) / (1000 m)
For the solid cylinder, let's substitute the values:
ρ2 = R * (0.785 cm^2) / (1000 m)
Here, R is the resistance, and it is the same for both rods because they are made of the same material.
Since R is common to both expressions, we can simplify the comparison of resistivity values as follows:
ρ1 / ρ2 = [(1 cm^2) / (1000 m)] / [(0.785 cm^2) / (1000 m)]
Now, calculating the above expression, we find:
ρ1 / ρ2 = 1.27
This ratio implies that the resistivity of the rectangular prism is approximately 1.27 times greater than the resistivity of the solid cylinder-shaped rod.
To answer your question, the rectangular prism with dimensions 1cm x 1cm x 1000m has a greater resistivity compared to the solid cylinder with a diameter of 1cm and a length of 1000m.
Your intuition about the importance of area in determining resistivity was correct. In this case, the rectangular prism has a larger cross-sectional area compared to the solid cylinder, resulting in a higher resistivity value.