To tackle the problem of finding the magnetic induction vector B for the scenarios you've described, we can indeed use Ampère's Law. This law states that the line integral of the magnetic field B around a closed loop is equal to the permeability of free space (μ₀) times the total current I enclosed by that loop. Let's break down each part of your question step by step.
Magnetic Induction of an Infinite Plane Carrying Current
For an infinite plane carrying a current with a linear density i, we can visualize the current as flowing uniformly across the entire plane. To apply Ampère's Law, we need to choose an appropriate Amperian loop. A good choice here is a rectangular loop that extends above and below the plane, parallel to the current flow.
Choosing the Amperian Loop
- Let’s say we take a rectangular loop with a height of 2h and width w, where the plane lies at z = 0.
- The top side of the loop is at z = h and the bottom side is at z = -h.
Now, we can calculate the magnetic field B at a distance z from the plane. Due to symmetry, the magnetic field will be uniform and directed parallel to the plane. We can assume it points in the positive x-direction (for example).
Applying Ampère's Law
According to Ampère's Law:
∮ B · dl = μ₀ I_enc
For our rectangular loop, the contributions to the line integral come only from the two horizontal sides (the top and bottom) because the vertical sides are perpendicular to B and thus contribute nothing. The current enclosed by our loop is the linear current density i multiplied by the width w of the loop:
I_enc = i * w
Now, the integral becomes:
B * w = μ₀ (i * w)
From this, we can solve for B:
B = μ₀ i
This indicates that the magnetic induction vector B has a magnitude of μ₀ i and is directed parallel to the plane, following the right-hand rule based on the direction of the current.
Magnetic Induction of Two Parallel Infinite Planes
Now, let’s consider the case of two parallel infinite planes carrying currents of linear densities i and -i. The first plane carries current in one direction, while the second carries current in the opposite direction. Again, we can use Ampère's Law with a similar Amperian loop.
Setting Up the Amperian Loop
We can use a rectangular loop that extends between the two planes. Let’s say the first plane is at z = 0 and the second at z = d. The loop will have height d and width w, similar to the previous case.
Calculating the Magnetic Field
In this scenario, the magnetic fields produced by each plane will interact. The plane at z = 0 produces a magnetic field of μ₀ i directed in one direction, while the plane at z = d produces a magnetic field of μ₀ (-i) directed in the opposite direction.
Using Ampère's Law again, we find the net magnetic field between the planes:
B_total = B_1 + B_2
Between the planes, the fields add up because they are in the same direction:
B_total = μ₀ i + μ₀ (-i) = 0
Outside the planes, the fields will cancel each other out as well, leading to:
B = μ₀ i (above the top plane)
B = -μ₀ i (below the bottom plane)
Thus, the magnetic induction vector B is zero in the region between the two planes, while outside, it has a magnitude of μ₀ i directed away from the planes.
Summary of Results
To summarize:
- For a single infinite plane with current density i, the magnetic induction is B = μ₀ i, directed parallel to the plane.
- For two parallel planes with current densities i and -i, the magnetic induction is B = 0 between the planes and B = μ₀ i outside the top plane and B = -μ₀ i outside the bottom plane.
This approach using Ampère's Law allows us to effectively determine the magnetic fields generated by current-carrying planes, leveraging symmetry and the properties of magnetic fields. If you have any further questions or need clarification on any part, feel free to ask!
