Two concentric, circular wire loops, of radii r1=20.0 cm and r2= 40.0 cm, are located in an x

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Two concentric, circular wire loops, of radii r1=20.0 cm...

Two concentric, circular wire loops, of radii $r_{1}=20.0 \mathrm{~cm}$ and $r_{2}=$ $40.0 \mathrm{~cm}$, are located in an $x y$ plane; each carries a clockwise current of 11.0 A (Fig. 28-26). (a) Find the magnitude of the net magnetic dipole moment of the system. (b) Repeat for reversed current in the inner loop.

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a measure of the strength and the orientation of a magnet or current loop and is defined as the product of the current (I) and the area (A) of the loop. It is a vector quantity, meaning it has both magnitude and direction. For a loop, the magnetic dipole moment is generally given by the formula ? = I × A, where A is the vector area perpendicular to the plane of the loop, with its direction determined by the right-hand rule.

Area of a Circular Loop

The area of a circular loop is a fundamental geometric concept and is calculated using the formula A = ?r², where r is the radius of the circle. In the context of magnetic dipole moments, this area is multiplied by the current in the loop to determine the dipole moment's magnitude. This calculation is crucial for evaluating the magnetic properties of circular current loops.

Right-Hand Rule

The right-hand rule is a mnemonic used in electromagnetism to determine the direction of the magnetic field relative to the direction of conventional current. For a current loop, if you curl the fingers of your right hand in the direction of the current flow, your thumb will point in the direction of the magnetic dipole moment (or the magnetic field inside the loop). This rule is essential for assigning the correct vector direction to the dipole moment, especially when multiple loops have currents flowing in different directions.

Superposition Principle

The superposition principle in electromagnetism states that the net magnetic dipole moment of a system of current loops is the vector sum of the individual dipole moments. This principle allows for analyzing complex systems by adding the contributions of each loop, taking into account both their magnitudes and directions. When currents are reversed or arranged differently, the superposition principle helps in determining how the individual moments combine to produce the final overall moment.

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