To understand the scenario involving two current loops positioned along the x-axis and y-axis, we need to delve into the principles of magnetic fields generated by current-carrying loops. The problem states that the ratio of the resultant magnetic field at a common point on the axis to the individual magnetic fields at that same point is 2:1. Let’s break this down step by step.

Magnetic Field Due to a Current Loop

A current loop generates a magnetic field that can be calculated using the Biot-Savart Law. For a circular loop of radius \( r \) carrying a current \( I \), the magnetic field \( B \) at a point along the axis of the loop (at a distance \( z \) from the center of the loop) is given by the formula:

B = \frac{{\mu_0 I r^2}}{{2 (r^2 + z^2)^{3/2}}}

Here, \( \mu_0 \) is the permeability of free space. This formula shows that the magnetic field strength decreases with distance from the loop.

Configuration of the Current Loops

In our case, we have two identical loops. One loop lies in the x-y plane (let's call it Loop A), and the other in the y-z plane (Loop B). The common point where we want to find the magnetic field is along the z-axis, at a distance \( z \) from the origin.

Calculating Individual Magnetic Fields

For Loop A (in the x-y plane), the magnetic field at point \( P \) on the z-axis can be calculated using the formula mentioned above. Similarly, for Loop B (in the y-z plane), the magnetic field at the same point \( P \) will also be calculated using the same formula, but the direction of the magnetic field will be different due to the orientation of the loops.

• Magnetic field from Loop A at point P: \( B_A \) directed along the z-axis.
• Magnetic field from Loop B at point P: \( B_B \) also directed along the z-axis.

Resultant Magnetic Field

Since both magnetic fields are in the same direction (along the z-axis), the resultant magnetic field \( B_R \) at point \( P \) can be expressed as:

B_R = B_A + B_B

Given that both loops are identical and carry the same current, we can denote \( B_A = B \) and \( B_B = B \). Thus, the resultant magnetic field becomes:

B_R = B + B = 2B

Finding the Ratio

Now, we can find the ratio of the resultant magnetic field to the individual magnetic field:

\(\frac{B_R}{B} = \frac{2B}{B} = 2\)

This indicates that the resultant magnetic field at the common point on the z-axis is twice the individual magnetic field produced by each loop at that point.

Final Thoughts

Thus, the ratio of the resultant magnetic field to the individual magnetic field at the common point is indeed 2:1. This example illustrates how the superposition principle applies to magnetic fields, allowing us to simply add the contributions from multiple sources when they are aligned in the same direction.