The rectangular loop of wire shown in Fig. P 27.77 has a mass of 0.15 g per centimeter of len

step 1 In this question we have mass per centimeter of length that is new equals to 0 .15 gram per cent

The rectangular loop of wire shown in Fig. P 27.77 has a...

The rectangular loop of wire shown in Fig. $\mathrm { P } 27.77$ has a mass of 0.15$\mathrm { g }$ per centimeter of length and is pivoted about side ab on a frictionless axis. The current in the wire is 8.2$\mathrm { A }$ in the direction shown. Find the magnitude and direction of the magnetic field parallel to the $y$ -axis that will cause the loop to swing up until cause the makes an angle of $30.0 ^ { \circ }$ with the yz-plane.

The rectangular loop of wire shown in Fig. $\mathrm { P } 27.77$
has a mass of 0.15$\mathrm { g }$ per centimeter of length and is pivoted
about side ab on a frictionless axis. The current in the wire is
8.2$\mathrm { A }$ in the direction shown.
Find the magnitude and direction
of the magnetic field parallel to
the $y$ -axis that will cause the
loop to swing up until cause the
makes an angle of $30.0 ^ { \circ }$ with the
yz-plane.

Key Concepts

Center of Mass and Gravitational Torque

For objects with distributed mass, the gravitational force effectively acts at the center of mass. When the loop swings upward, the gravitational force generates a torque about the pivot. Calculating this torque involves finding the position of the center of mass relative to the pivot, which is essential for setting up the equilibrium condition between gravitational and magnetic torques.

Magnetic Force on Current-Carrying Conductors

When a current-carrying conductor is placed in a magnetic field, it experiences a force described by the Lorentz force law. This force acts perpendicular to both the current direction and the magnetic field. In problems involving loops or segments of wire in magnetic fields, this principle is used to determine the forces acting on different parts of the conductor.

Magnetic Torque on Current Loops

A current loop in a magnetic field experiences a torque that tends to align the loop’s magnetic moment with the magnetic field. The torque is given by the cross product of the magnetic moment and the magnetic field. This concept is crucial for understanding how magnetic fields can cause rotational motion in current loops.

Equilibrium of Torques

In rotational dynamics, an object is in equilibrium when the net torque acting on it is zero. For a current loop pivoted on one side, achieving a specific angular orientation involves balancing the magnetic torque against other torques, such as gravitational torque. This balance determines the magnitude and direction of the magnetic field required to maintain the loop in a given position.

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