Problem 67 - University Physics with Modern Physics

Question

A straight piece of conducting wire with mass \(M\) and length \(L\) is placed on a friction- less incline tilted at an angle \(\theta\) from the horizontal (Fig. 27.61 ) There is a uniform, vertical magnetic field \(\overrightarrow{\boldsymbol{B}}\) at all points (produced by an arrangement of magnets not shown in the figure). To keep the wire from shiding down the incline, a voltage source is attached to the ends of the wire. When just the right amount of current flows through the wire, the wire remains at rest. Determine the magnitude and direction of the current in the wire that will cause the wire to remain at rest. Copy the figure and draw the direction of the current on your copy. In addition, show in a free-body diagram all the forces that act on the wire.

Step-by-Step Solution

Answer

Current magnitude is \( I = \frac{Mg \sin \theta}{LB} \) flowing up the incline, opposing gravity.

Step 1: Analyze Forces Acting on the Wire

Identify the forces acting on the wire: gravitational force, magnetic force, and the normal force. The gravitational force can be separated into two components: one parallel to the plane (down the incline) and one perpendicular to the plane.

Step 2: Apply Newton's First Law

Since the wire remains at rest, the net force along the incline must be zero. Set up the equation for the forces along the incline: the gravitational force component along the plane must be counteracted by the magnetic force due to the current.

Step 3: Determine Gravitational Force Component

The component of the gravitational force acting down the incline is given by \( F_{g, parallel} = Mg \sin \theta \), where \( M \) is the mass of the wire, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of the incline.

Step 4: Determine Magnetic Force

The magnetic force acting on the wire is given by \( F_{B} = ILB \cos \theta \), where \( I \) is the current through the wire, \( L \) is the length of the wire, \( B \) is the magnetic field strength, and \( \cos \theta \) is used as the direction of the current is perpendicular to \( \overrightarrow{B} \).

Step 5: Set Forces Equal to Solve for Current

Set the gravitational force component equal to the magnetic force: \( Mg \sin \theta = ILB \). Solve for the current \( I \): \( I = \frac{Mg \sin \theta}{LB} \).

Step 6: Determine Direction of Current

Using the right-hand rule, determine that the current must flow in a direction such that the magnetic force acts upwards along the incline, counteracting the gravitational component.

Key Concepts

Newton's First Law

Newton's First Law, also known as the law of inertia, tells us that an object will remain at rest or in motion unless acted upon by a net external force. In the context of the original exercise, this concept helps us understand why the wire, when subjected to magnetic and gravitational forces, remains stationary on an incline.

For the wire to stay at rest on the incline, the forces acting on it must be balanced. In other words, the net force acting along the incline must equal zero.

• The gravitational force has a component that acts down the incline.
• The magnetic force needs to exactly counteract this gravitational force component to maintain equilibrium.

By applying Newton's First Law, we require these forces to balance out, ensuring that the wire does not move. Setting up the equation based on this principle allows for the calculation of the current necessary to achieve this balance.

gravitational force component

When a force acts at an angle on an object, it can be useful to break it down into components. The gravitational force acting on the wire can be divided into two components:

• The component parallel to the incline, responsible for trying to pull the wire down the slope.
• The component perpendicular to the incline, which does not affect the motion along the incline but influences the normal force.

For the problem at hand, the gravitational force component parallel to the incline is particularly important because it must be counteracted by the magnetic force for the wire to remain stationary.

Mathematically, the parallel component of the gravitational force is given by: \[ F_{g, \text{parallel}} = Mg \sin \theta \]where:

• \( M \) is the mass of the wire.
• \( g \) is the acceleration due to gravity.
• \( \theta \) is the angle of the incline.

This shows why we're interested in the "\( \sin \theta \)" component: it's directly influencing the extent of the gravitational pull along the slope.

right-hand rule

The right-hand rule is a handy mnemonic for understanding the direction of force vectors in physics, particularly in the context of magnetic forces. This rule helps determine the direction of the magnetic force resulting from a current flowing through a wire in a magnetic field.

For the given exercise, after identifying the forces that must balance each other, the direction the current should take can be ascertained using this rule:

• Point your thumb in the direction of the wire's current.
• Let your fingers curl in the direction of the magnetic field lines.
• Your palm will then point in the direction of the resultant magnetic force.

By using the right-hand rule, we ensure that the current flows in a direction that allows the magnetic force to act upwards, countervailing the component of gravitational force pulling the wire down the incline. This ensures equilibrium is maintained, and the wire stays put on the slope.

free-body diagram

A free-body diagram is an essential tool in physics that helps in visualizing the forces acting on an object. In the context of the problem given, drawing a free-body diagram for the wire can clarify how the various forces interact.

This diagram typically includes:

• The gravitational force divided into components: one parallel and one perpendicular to the incline.
• The normal force perpendicular to the surface of the incline.
• The magnetic force applied due to the current flowing through the wire.

By illustrating these forces, a free-body diagram simplifies the understanding of the problem, providing a straightforward way to identify how the forces relate to one another and maintain equilibrium.

Such a visualization helps ascertain that the normal force balances the perpendicular gravitational force, while the magnetic force needs to counteract the parallel gravitational force. By checking this diagram, students can ensure they accurately set up their equations to solve for the required current magnitude and direction.