Magnetism And Matter

Step 1: Ampere's Circuital Law.
Ampere's circuital law states that the line integral of the magnetic field around a closed loop is proportional to the total current passing through the loop: where is the magnetic field, is an infinitesimal element of the loop, is the permeability of free space, and is the current enclosed by the loop.
Step 2: Choose a Circular Path.
To calculate the magnetic field around a straight current-carrying conductor, we choose a circular path centered on the conductor. The symmetry of the situation means the magnetic field will have the same magnitude at all points on the circular path and will be directed tangentially to the path. Thus, and are parallel, and the dot product simplifies to .
Step 3: Integral Form of Ampere's Law.
The line integral around the circular path becomes: where is the radius of the circle (the distance from the wire to the point where the magnetic field is being calculated).
Step 4: Applying Ampere’s Law.
According to Ampere's law, this integral is equal to , where is the current passing through the conductor. Therefore, we have:
Step 5: Solving for .
Solving for the magnetic field , we get:
Step 6: Conclusion.
Thus, the magnetic field at a distance from a long straight current-carrying conductor is:

Step 1: Magnetic Moment of a Coil.
The magnetic moment of a single coil is given by: where is the current, and is the area of the coil. For a circular coil of radius , the area . Thus, the magnetic moment of each coil is:
Step 2: Resultant Magnetic Moment.
Since the two coils are placed perpendicular to each other and carry the same current , the resultant magnetic moment is the vector sum of the individual magnetic moments. Since the angle between the two moments is 90°, the resultant magnetic moment is:
Step 3: Conclusion.
The resultant magnetic moment of the two coils is .