Figure 29-73 a shows a length of wire carrying a current i...

Figure $29-73 a$ shows a length of wire carrying a current $i$ and bent into a circular coil of one turn. In Fig. $29-$ 73$b$ the same length of wire has been bent to give a coil of two turns, each of half the original radius. (a) If $B_{a}$ and $B_{b}$ are the magnitudes of the magnetic fields at the centers of the two coils, what is the ratio $B_{b} / B_{a} ?$ (b) What is the ratio $\mu_{b} / \mu_{a}$ of the dipole moment magnitudes of the coils?

Figure $29-73 a$ shows a length of wire carrying a current $i$ and bent into
a circular coil of one turn. In Fig. $29-$
73$b$ the same length of wire has been
bent to give a coil of two turns, each of
half the original radius. (a) If $B_{a}$ and $B_{b}$
are the magnitudes of the magnetic fields at the centers of the two coils,
what is the ratio $B_{b} / B_{a} ?$ (b) What is the
ratio $\mu_{b} / \mu_{a}$ of the dipole moment magnitudes of the coils?

Key Concepts

Conservation of Wire Length in Coil Design

When a fixed length of wire is used to create coils, the radius of each turn is determined by how the total length is divided among the turns. This conservation principle means that altering the number of turns directly affects the dimensions of each turn, subsequently modifying the magnetic field and dipole moment. Understanding this constraint is key in comparing the electromagnetic properties of different coil configurations made from the same total length of wire.

Coil Geometry and Turn Number

The geometry of a coil, including its radius and the number of turns, plays a significant role in determining its magnetic properties. Altering the shape by changing the radius or the number of loops affects both the distribution of the magnetic field and the effective dipole moment. A reconfiguration that involves using the same total wire length but different turn counts will inherently change these properties due to the geometric dependencies in the formulas for magnetic field and dipole moment.

Magnetic Field of a Current Loop

This concept refers to the fact that a current-carrying loop produces a magnetic field, whose magnitude at the center can be calculated using principles from magnetostatics, notably through the Biot–Savart law. The field strength depends on the current and the geometry of the loop, particularly its radius, which inversely affects the magnitude of the magnetic field at the center.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that characterizes the overall strength and orientation of a current distribution’s magnetic field. For a loop of current, it is defined as the product of the current and the area enclosed by the loop, with multiple turns resulting in an additive effect. This quantity is crucial for understanding the interaction of the coil with external magnetic fields.

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