Two long, parallel wires carry current in the x-y plane. One wire carries 30 A to the left along the x-axis. The other carries 50 A to the right along a parallel line at y = 0.28 m. At what y-axis position in meters is the magnetic field equal to zero?
μ₀I₁/2πy= μ₀I₂/2π(0.28-y)
I₁(0.28-y)= I₂y
30(0.28-y)=50y
80y=8.4
y=8.4/80=0.105 m
It'swrong!!
To find the y-axis position where the magnetic field is equal to zero, we can use the Biot-Savart law, which relates the magnetic field to the current and the distance from the wire.
The Biot-Savart law for the magnetic field produced by a current-carrying wire is given by:
B = (μ₀ / 4π) * (I / r)
Where:
B is the magnetic field
μ₀ is the permeability of free space (constant, approximately 4π × 10^(-7) T•m/A)
I is the current
r is the distance from the wire
Since we have two wires, we need to calculate the magnetic field for each wire separately, and then find the y-axis position where the two magnetic fields cancel each other out (equal to zero).
Let's calculate the magnetic field due to each wire and find the y-axis position where they cancel out:
For the wire carrying 30 A to the left:
Using the Biot-Savart law, we have:
B₁ = (μ₀ / 4π) * (I₁ / r₁)
For the wire carrying 50 A to the right:
Similarly, we have:
B₂ = (μ₀ / 4π) * (I₂ / r₂)
To find the y-axis position where the magnetic field is zero, we need to solve for r₂:
B₁ = B₂
(μ₀ / 4π) * (I₁ / r₁) = (μ₀ / 4π) * (I₂ / r₂)
Simplifying and rearranging the equation:
I₁ / r₁ = I₂ / r₂
r₂ = (r₁ * I₂) / I₁
Since the wires are parallel, their y-coordinate remains the same. So, we can substitute r₁ = 0.28 m and solve for r₂:
r₂ = (0.28 * 50) / 30
r₂ = 0.4667 m
Therefore, the y-axis position where the magnetic field is equal to zero is 0.4667 meters.