two copper wires A and B are of the same length. If the diameter of A is 3 mm and its resistance is 3 ohms, what is the diameter of B if its resistance is 0.08 ohm ?
To solve this problem, we can use the equation for resistance of a wire:
R = (ρ * L) / A
where R is the resistance, ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
First, we need to find the resistivity ρ, which is a property of the material the wire is made of. For copper, the resistivity is approximately 1.68 x 10^-8 Ωm.
Next, we need to rearrange the formula to solve for the cross-sectional area A:
A = (ρ * L) / R
Given that wire A has a diameter of 3 mm and a resistance of 3 ohms, and wire B has a resistance of 0.08 ohms, we can substitute these values into the formula.
For wire A:
Diameter = 3 mm = 0.003 m
Resistance = 3 Ω
Using the formula A = (ρ * L) / R
A = (1.68 x 10^-8 Ωm * L) / 3 Ω
Now, since we are comparing wire A and wire B, we know that they have the same length L. Therefore, we can consider the lengths to be the same and cancel them out in the equation.
A = (1.68 x 10^-8 Ωm) / 3 Ω
Now, we can find the cross-sectional area A of wire A.
A ≈ 5.6 x 10^-9 m^2
Now, let's find the diameter of wire B using the same formula:
For wire B:
Resistance = 0.08 Ω
Using the formula A = (ρ * L) / R
A = (1.68 x 10^-8 Ωm * L) / 0.08 Ω
Again, canceling out the length L, we have:
A = (1.68 x 10^-8 Ωm) / 0.08 Ω
Now, we can substitute the cross-sectional area A of wire A into the equation to find the diameter of wire B.
(1.68 x 10^-8 Ωm) / 0.08 Ω = (π * (DiameterB/2)^2) / (π * (0.003/2)^2)
Simplifying the equation further:
1.68 x 10^-8 m / 0.08 = (DiameterB/2)^2 / (0.003/2)^2
1.68 x 10^-8 m / 0.08 = (DiameterB/2)^2 / (0.0015)^2
1.68 x 10^-8 m / 0.08 = (DiameterB/2)^2 / (0.0015)^2
Solving for DiameterB:
(DiameterB/2)^2 = (1.68 x 10^-8 m / 0.08) * (0.0015)^2
(DiameterB/2)^2 = 5.04 x 10^-16 m^3 / 0.1152
(DiameterB/2)^2 = 4.371 x 10^-15 m^3
Taking the square root of both sides to solve for DiameterB:
DiameterB/2 = √(4.371 x 10^-15 m^3)
DiameterB/2 ≈ 2.093 x 10^-8 m
Multiplying both sides by 2:
DiameterB ≈ 4.186 x 10^-8 m
Therefore, the diameter of wire B is approximately 4.186 x 10^-8 meters.