A student makes a short electromagnet by winding 300 turns...

A student makes a short electromagnet by winding 300 turns of wire around a wooden cylinder of diameter $d=5.0 \mathrm{~cm}$. The coil is connected to a battery producing a current of $4.0 \mathrm{~A}$ in the wire. (a) What is the magnitude of the magnetic dipole moment of this device? (b) At what axial distance $z \geqslant d$ will the magnetic field have the magnitude $5.0 \mu \mathrm{T}$ (approximately one-tenth that of Earth's magnetic field)?

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Key Concepts

Axial Magnetic Field of a Circular Loop

The magnetic field on the axis of a circular current loop is derived from the Biot–Savart Law and provides insight into how the field strength varies with distance from the coil’s center. For a single loop, the expression shows that the field decreases with the cube of the distance from the loop at distances much larger than the coil's radius, a concept that is important when determining the location where the field has a specific magnitude.

Coil Geometry and Area Calculation

Understanding the geometry of the coil is essential because the area of each turn directly affects the magnetic dipole moment. For a circular coil, the area is determined by the formula A = ?r², where r is the radius. This calculation connects the physical dimensions of the coil to its magnetic properties, linking the design of the electromagnet with its performance in generating a magnetic field.

Magnetic Dipole Moment

In electromagnetism, the magnetic dipole moment represents the strength and orientation of a magnetic source, such as a current loop or solenoid. It is calculated by multiplying the current by the area of the loop and the number of turns (? = N·I·A). This quantity is central to understanding how the device interacts with external magnetic fields and serves as an analogy to a tiny bar magnet.

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