In what two senses does a current loop behave like a magnetic dipole?

The Biot-Savart Law is a fundamental principle in electromagnetism that describes how electric currents produce magnetic fields. It is essential for understanding the behavior of a current loop, which behaves similarly to a magnetic dipole. According to this law, the magnetic field \textbf{B} at a point in space is directly proportional to the current I and inversely proportional to the distance r from the current carrying wire.

Mathematically, the Biot-Savart Law can be expressed as: \[ dB = \frac{\mu_0}{4\pi}\frac{Id\vec{s}\times\hat{r}}{r^2} \] where \( dB \) is the infinitesimal magnetic field produced by an infinitesimal segment of current \( Id\vec{s} \), \( \mu_0 \) is the permeability of free space, \( \hat{r} \) is the unit vector from the current segment to the point of interest, and \( r \) is the distance from the segment to this point.

Magnetic field lines are a visual tool used to represent the magnetic field produced by electric currents and magnetic dipoles. They give a direction to the magnetic field at every point in space, generally going from the north pole to the south pole outside of a magnet or current loop, and completing their path through the material or inside of the loop.

Magnetic field lines are a representation of the direction and strength where they are denser, the field is stronger. These lines do not intersect, and they provide an intuitive way to visualize the magnetic effects of a current loop or a magnetic dipole. The manner in which these lines form loops and spread out from the north to the south pole illustrates one way a current loop is akin to a magnetic dipole.

Magnetic moment is a vector quantity that represents the strength and orientation of a magnet's or a current loop's magnetic field. It is a key concept in discussing how a current loop behaves like a magnetic dipole. The magnetic moment \( \vec{m} \) of a current loop is given by the product of the current \( I \) and the area \( A \) of the loop, and is oriented perpendicular to the plane of the loop following the right-hand rule.

The formula can be written as \[ \vec{m} = I\cdot A\cdot \hat{n} \] where \( \hat{n} \) is the unit vector normal to the loop's plane. Both the current loop and a magnetic dipole have a magnetic moment, and this intrinsic property determines how they interact with external magnetic fields.

When a magnetic dipole or a current-carrying loop is placed within an external magnetic field, it experiences a torque. This torque tends to align the dipole's magnetic moment with the external field. The magnitude and direction of the torque can be calculated using the cross-product of the magnetic moment with the magnetic field.

The formula for the torque is \[ \tau = \vec{m} \times \vec{B} \] where \( \vec{m} \) is the magnetic moment and \( \vec{B} \) is the external magnetic field. The way both a current loop and a magnetic dipole experience torque in a magnetic field underlines their similar behaviors, satisfying the exercise's query and helping visualize the rotational dynamics due to magnetic interactions.