Figure 28-46 shows a wood cylinder of mass m=0.250 kg and length L=0.100 m, with N=10.0 turns of

step 1 Hi everyone, this is the problem based on motion of current carrying coil in uniform magnetic fi

Figure 28-46 shows a wood cylinder of mass m=0.250 kg and...

Figure $28-46$ shows a wood cylinder of mass $m=0.250 \mathrm{kg}$ and length $L=0.100 \mathrm{m},$ with $N=10.0$ turns of wire wrapped around it longitudinally, so that the plane of the wire coil contains the long central axis of the cylinder. The cylinder is released on a plane inclined at an angle $\theta$ to the horizontal, with the plane of the coil parallel to the incline plane. If there is a vertical uniform magnetic field of magnitude 0.500 $\mathrm{T}$ , what is the least current $i$ through the coil that keeps the cylinder from rolling down the plane?

Figure $28-46$ shows a wood cylinder of mass $m=0.250 \mathrm{kg}$ and length $L=0.100 \mathrm{m},$ with $N=10.0$ turns of wire wrapped around it longitudinally, so that the
plane of the wire coil contains the long central axis of the cylinder. The
cylinder is released on a plane inclined at an angle $\theta$ to the horizontal, with the plane of the coil parallel to the incline plane. If there is a vertical uniform magnetic field of magnitude
0.500 $\mathrm{T}$ , what is the least current $i$ through the coil that keeps the cylinder from rolling down the plane?

Key Concepts

Equilibrium of Torques

Equilibrium of torques occurs when the sum of all rotational forces (torques) acting on a system is zero. In problems where multiple torques—such as gravitational torque and magnetic torque—are present, setting the net torque to zero allows one to determine the conditions required for static equilibrium. This balance is key to solving problems that involve preventing motion due to competing influences.

Gravitational Force on an Inclined Plane

When an object is on an inclined plane, the component of gravitational force acting along the plane is proportional to the sine of the angle of the incline. This force tends to cause the object to accelerate or roll down the plane, and understanding it is crucial for analyzing equilibrium conditions in systems where gravity competes with other forces.

Magnetic Torque

Magnetic torque arises when a magnetic dipole is placed in an external magnetic field, resulting in a force that tends to align the dipole with the field. The magnitude of this torque is given by the cross product of the magnetic dipole moment and the magnetic field. This principle is used to counteract other torques (such as gravitational torque) in various physical systems.

Magnetic Dipole Moment

A magnetic dipole moment is a vector quantity that characterizes the magnetic effect of a current loop or coil. It is defined as the product of the current and the area of the loop, multiplied by the number of turns if the coil is multi-turn. This concept is essential in understanding how current-carrying loops interact with external magnetic fields to produce forces and torques.

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