Problem 34 - University Physics with Modern Physics

Question

An electromagnet produces a magnetic field of 0.550 T in a cylindrical region of radius 2.50 \(\mathrm{cm}\) between its poles. A straight wire carrying a current of 10.8 \(\mathrm{A}\) passes through the center of this region and is perpendicular to both the axis of the cylindrical region and the magnetic field. What magnitude of force is exerted on the wire?

Step-by-Step Solution

Answer

The force on the wire is 0.297 N.

Step 1: Understand the scenario

We have a magnetic field with a strength of 0.550 T in a circular region with a radius of 2.50 cm. A wire carrying a current of 10.8 A passes perpendicularly through this magnetic field. Our task is to find the force exerted on the wire.

Step 2: Determine the relevant formula

The force exerted on a current-carrying wire in a magnetic field is given by the formula: \[ F = I \times L \times B \times \sin(\theta) \]where \( F \) is the force, \( I \) is the current, \( L \) is the length of the wire in the magnetic field, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the direction of the current and the magnetic field. Since the wire is perpendicular to the magnetic field, \( \theta = 90^\circ \) and \( \sin( 90^\circ ) = 1 \).

Step 3: Calculate the length of wire in the field

Since the wire passes through the entire diameter of the cylindrical region, the length \( L \) is equal to the diameter of the circle. Given that the radius \( r = 2.50 \text{ cm} = 0.025 \text{ m} \), the diameter \( d = 2 \times 0.025 \text{ m} = 0.050 \text{ m} \).

Step 4: Calculate the force on the wire

Substitute the known values into the force equation. \[ F = 10.8 \text{ A} \times 0.050 \text{ m} \times 0.550 \text{ T} \times 1 = 0.297 \text{ N} \].

Step 5: Write the final solution

The magnitude of the force exerted on the wire is 0.297 N.

Key Concepts

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. Understanding magnetic fields is crucial in analyzing problems involving forces, especially in scenarios involving electromagnets and current-carrying wires. In the given problem, the magnetic field strength is 0.550 T (tesla), a measurement that indicates the intensity of the magnetic force in the region.

Magnetic fields are typically represented by lines that show the direction and strength of the field. The closer the lines, the stronger the field. In this case, the magnetic field is uniform in a cylindrical region, which means it has the same strength and direction throughout the region where it interacts with the wire. It's essential to know the direction of the magnetic field because it impacts how the force acts on a current-carrying wire, a concept informed by the right-hand rule.

• Strength of magnetic field: 0.550 T
• Cylindrical region: uniform field

Current-Carrying Wire

When a wire carries an electric current, a magnetic field forms around it. The current, which is the flow of electric charge, interacts with external magnetic fields present in the environment. In this problem, the current is flowing through the wire at 10.8 A (amperes). This current passes through the cylindrical magnetic field created by the electromagnet.

When the wire enters a magnetic field, it experiences a force due to the interaction between the wire's magnetic field and the external magnetic field. The direction of this force can be determined using the right-hand rule: Point your thumb in the direction of the current and your fingers in the direction of the magnetic field; your palm faces the direction of the force. Since the wire is perpendicular to the magnetic field in this problem, the interaction is maximal, making this a straightforward application of the force formula \[ F = I \times L \times B \times \sin(\theta) \] where \( \theta \) is 90 degrees, meaning \( \sin(90^\circ) = 1 \). The angle makes the force straightforward to compute.

Physics Problem Solving

Physics problem solving often involves breaking a problem into clear steps and applying known formulas to find an unknown quantity. Let's review the steps that helped find the force on the wire in this problem:

First, understand the scenario by identifying all given values and conditions. Knowing the geometry of the region (cylindrical and uniform magnetic field) and the specifics of the wire (perpendicular placement and current intensity) was crucial.

• Magnetic Field: 0.550 T
• Current: 10.8 A

• Step 2 involves selecting the correct formula for the force on a current-carrying wire: \[ F = I \times L \times B \times \sin(\theta) \] With \( \theta \) determined from the geometry as 90 degrees.
• Next, calculate the dimensions involved, such as the length of the wire in the field, which is equal to the diameter of the cylindrical region.
• Finally, substitute all known values into the formula to compute the force exerted on the wire, yielding the result of 0.297 N.

Each step builds on the previous one, highlighting the importance of having a structured method when solving physics problems.