Problem 69 - 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. P27.69). There is a uniform, vertical magnetic field \(\vec { B }\) at all points (produced by an arrangement of magnets not shown in the figure). To keep the wire from sliding 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

The current magnitude is \( I = \frac{Mg \sin \theta}{LB} \) and it should flow upwards along the incline.

Step 1: Understanding the Problem

The problem involves a conducting wire on an incline with a magnetic field that keeps it from sliding. The wire is subject to gravitational and magnetic forces. Our goal is to find the current that balances these forces.

Step 2: Identify Forces Acting on the Wire

We need to consider the gravitational force, which acts downward, and the magnetic force that acts perpendicular to the wire, as well as the incline of the plane. The wire remains at rest when these forces are balanced.

Step 3: Calculate Gravitational Force Component

The gravitational force acting along the incline is given by the component of weight parallel to the incline, i.e., \( F_{gravity} = Mg \sin \theta \) where \( g \) is the acceleration due to gravity.

Step 4: Apply Magnetic Force Equation

The magnetic force on the wire because of the current is given by \( F_{magnetic} = ILB \) where \( I \) is the current, \( L \) is the length of the wire, and \( B \) is the magnetic field strength.

Step 5: Equate Forces to Keep Wire at Rest

To prevent the wire from sliding, the magnetic force must equal the gravitational force component along the incline: \( ILB = Mg \sin \theta \). Solving this equation gives the current \( I = \frac{Mg \sin \theta}{LB} \).

Step 6: Determine Direction of Current

By using the right-hand rule, we deduce the direction of the current must be such that the force (created by the interaction of the current and magnetic field) acts upwards along the incline to counteract the gravitational pull. This means the current should flow upward if the B-field is vertically upwards.

Key Concepts

Current in a Conductor

Current in a conductor refers to the flow of electric charge through a material. In our scenario, it's crucial because the current flowing through the conducting wire generates a magnetic force. This magnetic force combats the wire's slide down the incline.
By applying Ohm's law and principles of magnetism, we can calculate the necessary current, ensuring it balances forces and keeps the wire stationary.
To solve the problem, recognize how the magnetic force generated by this current interacts with the magnetic field.

• The magnetic force depends directly on the current, size of the wire, and the strength of the magnetic field, summed in the equation: \( F_{magnetic} = ILB \).
• The direction of current affects which way the magnetic force pushes the wire.

By maintaining the correct current calculated by \( I = \frac{Mg \sin \theta}{LB} \), the wire balances perfectly with gravitational forces.

Incline Plane

An incline plane is a flat surface tilted at an angle relative to the horizontal. It's an essential element here as it complicates the forces acting on the wire. On an incline, gravity doesn't pull an object directly down; instead, it has two components:

• Parallel component to the surface, causing the wire to slide
• Perpendicular component pushing into the plane

For our scenario:
Gravitational force along the slope (\( F_{gravity} \)) is calculated as \( Mg \sin \theta \).
This component is vital because the magnetic force must match it for the wire to remain stationary.
The incline’s angle determines how significant the sliding force is, with steeper inclines requiring more current to balance.

Magnetic Field Direction

The magnetic field direction is a key concept impacting the force the wire experiences. When a current flows in a conductor within a magnetic field, a force is exerted. This phenomenon, known as the motor effect, depends on both current direction and magnetic field.
The right-hand rule helps determine force direction: point your thumb in the current's flow, and your fingers in the magnetic field's direction, then your palm faces the resulting force direction.
In this exercise:

• The magnetic field is vertical; assume an upward direction for simplicity.
• The force direction needs to counteract gravity's pull down the incline.

Thus, adjusting the direction of the current to align with the magnetic field's effect is crucial for balancing forces.