An electric motor contains a 250 -turn circular coil 5.4 cm in diameter. If it develops a maxim

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An electric motor contains a 250 -turn circular coil 5.4 cm...

An electric motor contains a 250 -turn circular coil $5.4 \mathrm{~cm}$ in diameter. If it develops a maximum torque of $2.0 \mathrm{~N} \cdot \mathrm{m}$ at a current of $3.1 \mathrm{~A}$, what's the magnetic field strength?

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a measure of the strength and orientation of a magnetic source. For a current-carrying loop, it is defined as the product of the current, the number of turns, and the area of the loop. This concept is vital for understanding how current loops interact with external magnetic fields.

Torque on a Current Loop

When a current loop is placed in a magnetic field, it experiences a torque that tends to align its magnetic dipole moment with the magnetic field. The torque is given by the cross product of the magnetic dipole moment and the magnetic field, reaching a maximum when the dipole moment is perpendicular to the field.

Area of a Circle

The area of a circle is a basic geometric concept used to calculate the area of a circular loop, which is necessary for determining the magnetic dipole moment. The area is computed using the formula A = ?r², where r is the radius of the circle.

Relationship between Torque, Magnetic Dipole Moment, and Magnetic Field

In the context of magnetic interactions, the torque experienced by a current loop can be expressed as ? = ?B sin?, where ? is the magnetic dipole moment, B is the magnetic field strength, and ? is the angle between them. For maximum torque, sin? equals 1, which simplifies the relationship to ? = ?B, allowing for the determination of any one quantity if the others are known.

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