The rectangular loop shown in Fig. P27.78 is pivoted about...

The rectangular loop shown in Fig. P27.78 is pivoted about the $y$ -axis and carries a current of 15.0$\mathrm { A }$ in the direction indicated. (a) If the loop is in a uniform magnetic field with magnitude 0.48 T in the $+ x$ - direction, find the magnitude and direction of the torque required to hold the loop in the position shown. (b) Repeat part (a) for the case in which the field is in the $- z$ -direction. (c) For each of the above magnetic fields, what torque would be required if the loop were pivoted about an axis through its center, parallel to the y-axis?

The rectangular loop shown in Fig. P27.78 is pivoted
about the $y$ -axis and carries a
current of 15.0$\mathrm { A }$ in the direction
indicated. (a) If the loop is in a uniform magnetic field with
magnitude 0.48 T in the $+ x$ -
direction, find the magnitude and
direction of the torque required
to hold the loop in the position shown. (b) Repeat part (a) for the
case in which the field is in the
$- z$ -direction. (c) For each of
the above magnetic fields, what
torque would be required if the
loop were pivoted about an axis
through its center, parallel to the
y-axis?

Key Concepts

Pivot Axis and Moment Arm Consideration

When calculating the torque required to hold a current loop in position, the location of the pivot axis is important. The effective moment arm (or lever arm) may change based on whether the loop rotates about its center or an edge. This difference affects the net torque needed to balance the magnetic forces acting on different parts of the loop, making it a crucial consideration for accurately determining the rotational dynamics of the system.

Uniform Magnetic Field

A uniform magnetic field is one in which the field strength and direction remain constant throughout the region of interest. This simplification allows for straightforward calculations of the forces and torques acting on current loops and other magnetic dipoles, since the magnetic interaction is the same at every point within the field.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that characterizes the strength and orientation of a current-carrying loop. It is defined as the product of the current and the area of the loop, with a direction given by the right?hand rule. This concept is central to understanding how current loops interact with magnetic fields, determining the magnitude and direction of the generated torque when placed in an external field.

Torque on a Magnetic Dipole

The torque experienced by a current loop in a magnetic field is given by the cross product of the magnetic dipole moment and the magnetic field, expressed as ? = ? × B. This principle explains how the loop tends to rotate to align its dipole moment with the magnetic field. The magnitude of this torque is dependent on the sizes of both the magnetic moment and the magnetic field, as well as the sine of the angle between them.

Transcript

Please give Ace some feedback