Problem 36 - University Physics with Modern Physics

Question

An electromagnet produces a magnetic field of 0.550 T in a cylindrical region of radius 2.50 cm between its poles. A straight wire carrying a current of 10.8 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 does this field exert on the wire?

Step-by-Step Solution

Answer

The force exerted on the wire is 0.297 N.

Step 1: Identify the Given Values

We are given the magnetic field strength \( B = 0.550 \text{ T} \), the radius of the cylindrical region \( r = 2.50 \text{ cm} \), and the current in the wire \( I = 10.8 \text{ A} \). We need to calculate the force exerted on the wire.

Step 2: Recall the Formula for Magnetic Force

The force exerted on a wire carrying a current in a magnetic field is given by the formula \( F = I \cdot L \cdot B \cdot \sin \theta \), where \( I \) 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 current direction and the magnetic field. Since the wire is perpendicular to the magnetic field, \( \theta = 90^\circ \) and \( \sin \theta = 1 \).

Step 3: Determine the Length of the Wire in the Magnetic Field

Since the wire passes through the diameter of the cylindrical region, the length \( L \) of the wire in the magnetic field equals the diameter of the cylinder, which is twice the radius: \( L = 2 \cdot r = 2 \times 2.50 \text{ cm} = 5.00 \text{ cm} = 0.0500 \text{ m} \).

Step 4: Calculate the Force on the Wire

Substitute the known values into the formula: \( F = 10.8 \text{ A} \times 0.0500 \text{ m} \times 0.550 \text{ T} \times 1 = 0.297 \text{ N} \). Thus, the magnitude of the force exerted on the wire is 0.297 N.

Key Concepts

Electromagnetism

Electromagnetism is a fundamental force of nature, encompassing both electric and magnetic phenomena. It describes how electric charges interact with each other and with magnetic fields. One of the most fascinating applications of electromagnetism is the concept of an electromagnet, which involves a coil of wire that generates a magnetic field when an electric current flows through it. This field can be used for various applications, from lifting heavy metal objects to driving electrical motors.
In the context of our exercise, the magnetic field is created within a cylindrical region by an electromagnet. This is a controlled area where the magnetic effects are harnessed and measured. Understanding electromagnetism, in this case, helps us see how electric currents and magnetic fields can exert forces and influence other materials within the field's reach.

Magnetic Field Strength

The magnetic field strength is a measure of the intensity or strength of a magnetic field. It is represented by the symbol \( B \) and typically measured in teslas (T). This value helps us understand how strong the magnetic field is and how it will interact with other materials, such as a current-carrying wire, in its vicinity.
In the original problem, the magnetic field strength is given as 0.550 T. This indicates a relatively strong field, capable of exerting a noticeable force on any conductive material within its range. The magnetic field strength is crucial in determining the magnitude of the magnetic force exerted on the wire, as shown in the relevant formula \( F = I \cdot L \cdot B \cdot \sin \theta \).

• \( B \) affects how much the field can influence the wire.
• The greater the magnetic field strength, the stronger the force on the wire.

The angle \( \theta \) plays a role too. In this scenario, being 90 degrees maximizes the effect of \( B \).

Current-Carrying Wire

A current-carrying wire is a fundamental element in the study of electromagnetism. When an electric current \( I \) flows through a wire, it generates a magnetic field around it. This interaction is at the core of how devices such as generators and motors work.
In our exercise, the wire carries a current of 10.8 A and is placed within a magnetic field. This setup is key for observing the magnetic force in action. The direction of the current relative to the magnetic field affects the force pattern:

• Perpendicular alignment of the wire and field maximizes the force.
• Magnetic forces follow the right-hand rule; pointing in a direction perpendicular to both the current and the magnetic field direction.

The length \( L \) of the wire in the magnetic field plays a critical role too. It determines how much of the wire is affected and thus influences the total force calculated using \( F = I \cdot L \cdot B \cdot \sin \theta \). Here, the wire's length is the diameter of the cylindrical region, which shows how distance in the field impacts overall force exerted on the wire.