Two vertical parallel conductors X and Y are 0.12m apart and carry currents of 2A and 4A respectively in a downward direction.
Draw the resultant flux pattern between X and Y. ignoring the earth's magnetic field, find the distance from X of a point where the magnetic fields due to X and Y neutralize each other?
To draw the resultant flux pattern between the vertical parallel conductors X and Y, you can use Ampere's right-hand rule and consider the direction of the magnetic field around each conductor.
Using Ampere's right-hand rule:
1. Take your right hand and curl your fingers in the direction of the current flow.
2. For conductor X, point your thumb in the direction of the current flow (downward). The field lines around conductor X will be counterclockwise.
3. For conductor Y, point your thumb in the direction of the current flow (downward). The field lines around conductor Y will be counterclockwise but stronger than those around conductor X because the current in Y is twice the current in X.
Now, considering the resultant flux pattern, the magnetic field lines will wrap around each conductor individually but will also interact with each other in the region between the two conductors. The resultant pattern will have concentric circular magnetic field lines around each conductor, getting progressively weaker as you move away from the conductors.
To find the distance from conductor X where the magnetic fields due to X and Y neutralize each other, you need to consider the magnetic field at a point between the conductors.
Let's assume that the distance from conductor X to the neutralization point is d. At this point, the magnetic fields due to X and Y will be equal in magnitude but opposite in direction. Mathematically, we can determine this by equating their magnitudes:
(B_x) = (B_y)
Using the formula for the magnetic field produced by a long straight conductor:
(B_x) = (μ₀ * I_x) / (2π * d)
(B_y) = (μ₀ * I_y) / (2π * (0.12 - d))
Where:
(B_x) = Magnetic field due to conductor X
(B_y) = Magnetic field due to conductor Y
μ₀ = Permeability of free space (constant)
I_x = Current in conductor X
I_y = Current in conductor Y
d = Distance from conductor X to the neutralization point (what we want to find)
Now, set (B_x) equal to (B_y):
(μ₀ * I_x) / (2π * d) = (μ₀ * I_y) / (2π * (0.12 - d))
Simplifying this equation, you'll find:
2 * d = 0.12 - d
3 * d = 0.12
d = 0.04 meters
Therefore, the distance from conductor X to the point where the magnetic fields due to X and Y neutralize each other is 0.04 meters.