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

step 1 Here we're considering a wooden cylinder on an inclined plane where there is 10 loops of wire wr

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 $\underline{\imath}$ 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 $\underline{\imath}$ through the coil that keeps the cylinder from rolling down the plane?

Key Concepts

Lorentz Force

This is the force exerted on a current-carrying conductor when placed in a magnetic field, given by the cross product between the current’s direction and the magnetic field vector. It is a fundamental concept in electromagnetism, and its magnitude and direction determine how magnetic fields interact with moving charges in conductors.

Magnetic Torque on Current Loops

When a loop or coil carrying a current is placed in a magnetic field, it experiences a torque due to the magnetic forces acting on different segments of the loop. This torque tends to rotate the loop and is the basis for many applications such as electric motors. The magnitude of the torque depends on parameters like current, the number of turns, the area of the loop, and the angle between the magnetic moment and the field.

Equilibrium Condition

In mechanics, equilibrium is reached when the net force and net torque acting on a system are zero. This concept is essential when determining the conditions under which a system remains stationary or moves with a constant velocity. Balancing forces such as gravitational pull and magnetic forces can lead to static equilibrium even when multiple forces are acting simultaneously.

Gravitational Component on an Inclined Plane

When an object is placed on an inclined plane, the gravitational force can be resolved into two components: one perpendicular to the plane and one parallel to it. The parallel component is responsible for trying to slide the object down the incline, and understanding this decomposition is crucial for analyzing and balancing forces in problems that combine gravity with other effects like magnetic forces.

Transcript

Please give Ace some feedback