6. A square aluminum rod is 1.0 meters long and 5.0 mm on edge. What is the resistance between its ends? What must be the diameter of a circular 1.0 meter long copper rod if its resistance is to be the same
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To find the resistance of the square aluminum rod, we can use the formula:
Resistance = Resistivity * (Length / Cross-sectional area)
First, we need to determine the cross-sectional area of the square aluminum rod.
Given that the edge length of the square aluminum rod is 5.0 mm, we can calculate its cross-sectional area (A) using the formula for the area of a square:
A = (edge length)^2
A = (5.0 mm)^2
A = 25 mm^2
Now, we need to convert the cross-sectional area from mm^2 to m^2 since the length is given in meters. We can achieve this by dividing the area by 1,000,000 (since there are 1,000,000 mm^2 in 1 m^2):
A = 25 mm^2 / 1,000,000
A = 0.000025 m^2
Next, we need to determine the resistivity (ρ) of aluminum. The resistivity of aluminum is approximately 2.82 x 10^(-8) ohm-meters.
Using the given length of the square aluminum rod (1.0 meter), the calculated cross-sectional area (0.000025 m^2), and the resistivity of aluminum (2.82 x 10^(-8) ohm-meters), we can calculate the resistance (R) using the formula mentioned earlier:
R = Resistivity * (Length / Cross-sectional area)
R = 2.82 x 10^(-8) ohm-meters * (1.0 meter / 0.000025 m^2)
R = 2.82 x 10^(-8) ohm-meters * 40,000 ohms
Calculating the above expression will give us the resistance of the square aluminum rod in ohms.
To find the diameter of the circular copper rod that would have the same resistance, we need to consider the resistivity of copper (which is approximately 1.68 x 10^(-8) ohm-meters).
Using the given length of 1.0 meter and the resistivity of copper (1.68 x 10^(-8) ohm-meters), we can calculate the cross-sectional area (A) of the circular copper rod required to have the same resistance as the square aluminum rod:
A = Resistance / (Resistivity * Length)
Substituting the resistance of the square aluminum rod (calculated earlier) and the resistivity of copper into the formula above will give us the desired cross-sectional area of the copper rod.
Finally, using the formula for the area of a circle:
A = π * (diameter/2)^2
We can rearrange the formula to solve for the diameter (d):
diameter = sqrt(4 * A / π)
Substituting the calculated cross-sectional area of the copper rod into the formula above will give us the required diameter of the circular copper rod in meters.