Figure 28-49 shows a current loop A B C D E F A carrying a...

Figure $28-49$ shows a current loop $A B C D E F A$ carrying a current $i=$ 5.00 A. The sides of the loop are parallel to the coordinate axes shown, with $A B=20.0 \mathrm{cm}, B C=30.0 \mathrm{cm},$ and $F A=$ 10.0 $\mathrm{cm} .$ In unit-vector notation, what is the magnctic dipole moment of this loop? (Hint: Imagine equal and opposite currents $i$ in the line segment $A D$ ; then treat the two rectangular loops $A B C D A$ and $A D E F A . )$

Figure $28-49$ shows a current loop $A B C D E F A$ carrying a current $i=$
5.00 A. The sides of the loop are parallel
to the coordinate axes shown, with
$A B=20.0 \mathrm{cm}, B C=30.0 \mathrm{cm},$ and $F A=$ 10.0 $\mathrm{cm} .$ In unit-vector notation, what is the magnctic dipole moment of this
loop? (Hint: Imagine equal and opposite currents $i$ in the line segment $A D$ ;
then treat the two rectangular loops
$A B C D A$ and $A D E F A . )$

Key Concepts

Superposition Principle in Loop Decomposition

For complex current loop geometries, it can be useful to decompose the loop into simpler segments or subloops whose contributions to the overall magnetic dipole moment can be calculated individually and then combined. This method relies on the superposition principle, which states that the total effect is the sum of the individual effects, and is especially effective when dealing with circuits or loops that are combinations of simpler shapes.

Right-Hand Rule

The right-hand rule is a practical tool used to determine the direction of vector quantities in electromagnetism, such as the magnetic dipole moment and the magnetic field around a current-carrying wire. By curling the fingers of the right hand in the direction of the current flow in a loop, the thumb points in the direction of the area vector (and thus the magnetic dipole moment). This rule ensures consistency in the orientation of vectors throughout electromagnetic analysis.

Area Vector

The area vector associated with a planar current loop represents both the magnitude of the area and its orientation. Its magnitude is equal to the area of the loop, and its direction is perpendicular to the plane of the loop, determined by applying the right-hand rule to the current flow. In many electromagnetic problems, the area vector is used to express quantities such as the magnetic dipole moment in a compact, vector form.

Magnetic Dipole Moment

The magnetic dipole moment of a current loop is a vector quantity that characterizes the strength and orientation of the magnetic source. It is defined as the product of the current and the area enclosed by the loop, with its direction determined by the right?hand rule. This concept is fundamental in electromagnetism as it describes how current distributions generate magnetic fields, and it is used in analyzing the interaction of magnetic fields with materials or devices that have intrinsic magnetic moments.

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