To analyze the situation with the four coaxial cylindrical conductors, we need to consider the currents flowing through them and how they interact with each other. The conductors A, B, C, and D have radii R, 2R, 3R, and 4R, respectively. Conductors A and C carry currents I and 3I into the plane of the paper, while conductors B and D carry currents 4I and 2I outward from the plane of the paper. Our goal is to determine the net magnetic field at a point in space due to these currents.
Understanding Magnetic Fields Around Conductors
When current flows through a conductor, it generates a magnetic field around it. The direction of this magnetic field can be determined using the right-hand rule: if you point your thumb in the direction of the current, your fingers curl in the direction of the magnetic field lines. For coaxial conductors, the magnetic fields produced by each conductor can be calculated using Ampère's Law.
Applying Ampère's Law
Ampère's 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:
- For conductor A (radius R): The current is I. The magnetic field at a distance r from the center (where r < R) is zero, and for r > R, it can be calculated as:
- B_A = (μ₀ * I) / (2πr)
- For conductor B (radius 2R): The current is 4I. For r < 2R, the magnetic field is zero, and for r > 2R:
- B_B = (μ₀ * 4I) / (2πr)
- For conductor C (radius 3R): The current is 3I. For r < 3R:
- B_C = (μ₀ * 3I) / (2πr)
- For conductor D (radius 4R): The current is 2I. For r < 4R:
- B_D = (μ₀ * 2I) / (2πr)
Calculating the Net Magnetic Field
To find the net magnetic field at a specific point, we need to consider the contributions from each conductor based on their distances from that point. The magnetic fields produced by currents flowing in opposite directions will subtract from each other, while those flowing in the same direction will add together.
Let's assume we are interested in a point at a distance greater than 4R (i.e., r > 4R). The net magnetic field B_net can be expressed as:
- B_net = B_A + B_C - B_B - B_D
Substituting the expressions we derived earlier:
- B_net = (μ₀ * I) / (2πr) + (μ₀ * 3I) / (2πr) - (μ₀ * 4I) / (2πr) - (μ₀ * 2I) / (2πr)
This simplifies to:
- B_net = (μ₀ / (2πr)) * (I + 3I - 4I - 2I)
Calculating the terms inside the parentheses gives:
- B_net = (μ₀ / (2πr)) * (-2I)
Final Result
Thus, the net magnetic field at a point outside the outermost conductor (at distance r > 4R) is:
- B_net = - (μ₀ * 2I) / (2πr)
This indicates that the magnetic field is directed opposite to the direction of the current flowing in the conductors. The negative sign signifies the direction of the magnetic field relative to the chosen coordinate system. Understanding these interactions helps us grasp the fundamental principles of electromagnetism and the behavior of magnetic fields around current-carrying conductors.
