91Ó°ÊÓ

When a circular loop of wire carrying an electric current is placed in a magnetic field, it experiences a torque. The torque \( \vec{\tau} \) is the rotational force that the magnetic field exerts on the loop. It can be calculated using the formula:

• \( \vec{\tau} = \vec{\mu} \times \vec{B} \)

Here, \( \vec{\mu} \) is the magnetic dipole moment, and \( \vec{B} \) is the magnetic field vector. The cross product \( \times \) means that \( \vec{\tau} \) will be perpendicular to both \( \vec{\mu} \) and \( \vec{B} \).

In our exercise, we find the torque by calculating the cross product of the magnetic dipole moment \( \vec{\mu} = (0.00241\hat{i} - 0.00322\hat{j}) \) and the magnetic field \( \vec{B} = (0.50\hat{i} + 0.20\hat{k}) \). The resulting torque is \( \vec{\tau} = 0.000482 \hat{i} + 0.000644 \hat{k} \, \text{N} \cdot \text{m} \).

This torque determines how the loop will attempt to align itself within the magnetic field, rotating in the direction of the torque until equilibrium is reached.

A circular loop of wire is a common configuration in electromagnetism due to its symmetry and simplicity in analysis. The characteristics of a circular loop allow us to study fundamental electromagnetic concepts like magnetic dipole moment and induced currents. For this exercise, we'll consider the circular loop of wire that has a 8.0 cm (radius).

The loop carries a steady current, which gives rise to a magnetic dipole moment. The dipole moment \( \vec{\mu} \) is a vector that reflects the magnetic strength and orientation of the loop. It's given by the product of the current \( I \), the area \( A \) of the loop, and the direction as specified by a unit vector.

The area \( A \) of this circular loop is computed using the formula \( A = \pi r^2 \), which results in approximately 0.0201 m² for a radius of 8.0 cm. This area is crucial in determining the magnitude of the magnetic dipole moment, showcasing the direct relationship between the loop's size and its magnetic properties.

Magnetic fields exert forces on moving charges and magnetic materials within them. In this exercise, we have a uniform magnetic field represented by the vector \( \vec{B} = (0.50 \mathrm{~T}) \hat{\mathrm{i}} + (0.20 \mathrm{~T}) \hat{\mathrm{k}} \).

The magnetic field is a vector field around a magnet where magnetic forces can be observed. It is defined by both magnitude and direction, thus influencing the torque on a loop of wire. The strength of a magnetic field is measured in Tesla (T).

• The component \( 0.50 \mathrm{~T} \hat{\mathrm{i}} \) indicates the field's strength along the x-direction.
• The component \( 0.20 \mathrm{~T} \hat{\mathrm{k}} \) indicates the field's strength along the z-direction.

Understanding these components helps in comprehensively determining how a magnetic loop will react within the field, calculating both the torque and potential energy interactions. Here, the torque calculated gives an insight into how the loop will behave and align due to the imposed magnetic field.