A cylindrical copper cable carries a current of 1070 A. There is a potential difference of 1.6 10-2 V between two points on the cable that are 0.22 m apart. What is the radius of the cable?
To find the radius of the copper cable, we can use the formula for resistance and rearrange it to solve for the radius. The resistance of a cylindrical conductor can be calculated using the formula:
R = (ρ * L) / (A)
where:
R is the resistance of the conductor,
ρ is the resistivity of the material (copper in this case),
L is the length of the conductor, and
A is the cross-sectional area of the conductor.
In this case, we are given the current (I = 1070 A), the potential difference (V = 1.6 * 10^(-2) V), and the distance between the two points on the cable (d = 0.22 m).
First, we need to calculate the resistance using Ohm's Law:
R = V / I
Now, we can rearrange the resistance formula above to solve for the cross-sectional area (A):
A = (ρ * L) / R
The resistivity of copper (ρ) is approximately 1.7 * 10^(-8) Ω·m. The length (L) is the distance between the two points, which is given as 0.22 m.
Now, let's plug in the values and solve for A:
A = (1.7 * 10^(-8) Ω·m * 0.22 m) / (1.6 * 10^(-2) V / 1070 A)
Simplifying the expression:
A = (1.7 * 10^(-8) * 0.22) / (1.6 * 10^(-2) / 1070)
A = (3.74 * 10^(-9)) / (1.6 * 10^(-2) / 1070)
A = (3.74 * 10^(-9)) / (1.6 * 10^(-2)) * (1070)
A = (3.74 * 10^(-9) * 1070) / (1.6 * 10^(-2))
A ≈ 0.2504 m^2
The cross-sectional area of the copper cable is approximately 0.2504 square meters.
Next, we can use the formula for the area of a circle to find the radius:
A = π * r^2
Rearranging the formula:
r = sqrt(A / π)
Substituting the value of A:
r = sqrt(0.2504 / π)
Calculating the radius:
r ≈ sqrt(0.0797)
r ≈ 0.282 m
Therefore, the radius of the copper cable is approximately 0.282 meters.