A long horizontal wire carries a current of 48A . A second wire, made of 2.6mm diameter copper wire and parallel to the first, is kept in suspension magnetically 15cm below(Figure 1) .
Determine the magnitude of the current in the lower wire.
To determine the magnitude of the current in the lower wire, we can use Ampere's law. Ampere's law states that the magnetic field created by a current-carrying wire at a particular distance is directly proportional to the current flowing through the wire.
Given that the first wire is carrying a current of 48A, we need to calculate the magnetic field created by this wire at the position of the second wire. We can then use this magnetic field value to find the current in the lower wire.
To calculate the magnetic field created by the first wire at the position of the second wire, we can use the formula for the magnetic field due to a long straight wire:
B = (μ₀ * I) / (2π * r)
Where:
B is the magnetic field
μ₀ is the permeability of free space (constant value)
I is the current flowing through the wire
r is the distance from the wire
In this case, the distance from the first wire to the second wire is 15 cm, which is equivalent to 0.15 meters. The permeability of free space (μ₀) is a constant value of 4π * 10^(-7) Tm/A.
Substituting these values:
B = (4π * 10^(-7) Tm/A * 48A) / (2π * 0.15m)
Simplifying:
B = (8π * 10^(-7) T) / 0.3
B ≈ 8.377 x 10^(-6) T
Now that we have the magnetic field, we can use it to find the current in the lower wire. We know that the magnetic field (B) created by the first wire is equal to the magnetic field (B') created by the second wire:
B' = (μ₀ * I') / (2π * r)
Where I' is the current flowing through the lower wire. We can rearrange this equation to solve for I':
I' = (B' * (2π * r)) / μ₀
Substituting the values:
I' = (8.377 x 10^(-6) T * (2π * 0.15m)) / (4π * 10^(-7) Tm/A)
Simplifying:
I' ≈ 8.377 x 10^(-6) T * (2 * 0.15) / 4 x 10^(-7) A
I' ≈ 2.511 x 10^(-5) A
Therefore, the magnitude of the current in the lower wire is approximately 2.511 x 10^(-5) A.