A circular coil with 200 turns has a radius of 2.0 cm. (a)...

A circular coil with 200 turns has a radius of 2.0 cm. (a) What current through the coil results in a magnetic dipole moment of $3.0 \mathrm{Am}^{2}$ ? (b) What is the maximum torque that the coil will experience in a uniform field of strength $5.0 \times 10^{-2} \mathrm{T} ?$ (c) If the angle between $\mu$ and $B$ is $45^{\circ},$ what is the magnitude of the torque on the coil? (d) What is the magnetic potential energy of coil for this orientation?

A circular coil with 200 turns has a radius of 2.0 cm. (a) What current through the coil results in a magnetic dipole moment of $3.0 \mathrm{Am}^{2}$ ? (b) What is the maximum torque that the coil will experience in a uniform field of strength $5.0 \times 10^{-2} \mathrm{T} ?$ (c) If the angle between $\mu$ and $B$ is $45^{\circ},$ what is the magnitude of the torque on the coil?
(d) What is the magnetic potential energy of coil for this orientation?

Key Concepts

Magnetic Potential Energy

Magnetic potential energy is the energy associated with a magnetic dipole in an external magnetic field. It depends on the orientation of the dipole relative to the field and is expressed as the negative dot product of the magnetic dipole moment and the magnetic field. This energy reaches a minimum when the dipole is aligned with the field, which is a key principle in many magnetic phenomena.

Geometric Properties of a Circular Coil

The calculation of the area of a circular coil, which is derived from its radius, is essential in determining the magnetic dipole moment. In a coil, the area directly affects how much magnetic dipole moment is produced for a given current and number of turns, linking geometric factors to magnetic properties.

Magnetic Dipole Moment

This concept refers to the measure of the magnetic strength and orientation of a magnetic source, such as a current-carrying coil. It is calculated as the product of the current through the coil, the number of turns, and the area of the coil, and it is a fundamental parameter in determining how the coil interacts with external magnetic fields.

Torque on a Magnetic Dipole

Torque on a magnetic dipole placed in a magnetic field quantifies the twisting force acting on the dipole. It is given by the vector cross product of the magnetic dipole moment and the magnetic field, and its magnitude depends on the sine of the angle between them. This concept is crucial for understanding how magnetic dipoles align with magnetic fields.

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