Problem 38 - University Physics with Modern Physics

Question

An electromagnet produces a magnetic field of 0.550 \(\mathrm{T}\) in a cylindrical region of radius 2.50 \(\mathrm{cm}\) 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 is exerted on the wire?

Step-by-Step Solution

Answer

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

Step 1: Identify the Relevant Formula

The force exerted on a current-carrying wire in a magnetic field is given by the formula \( F = I L B \sin \theta \), 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 wire and the magnetic field. In this scenario, \( \theta = 90^\circ \), so \( \sin \theta = 1 \).

Step 2: Determine the Wire Length Affected

Since the wire passes through the center of the cylindrical region, and the force is only exerted within this region, we consider the diameter of the region as the length of the wire affected by the magnetic field. The diameter \( L \) of the region is twice the radius: \( L = 2 \times 2.50 \text{ cm} = 5.00 \text{ cm} = 0.050 \text{ m} \).

Step 3: Substitute Known Values into the Formula

Now that we have all the values, we substitute them into the formula: \( F = 10.8 \text{ A} \times 0.050 \text{ m} \times 0.550 \text{ T} \times 1 = 0.297 \text{ N} \).

Step 4: Calculate the Magnitude of the Force

Perform the multiplication to find the force: \( F = 0.297 \text{ N} \). This is the magnitude of the force exerted on the wire.

Key Concepts

Electromagnetism

Electromagnetism is a fundamental force of nature that encompasses the forces of electricity and magnetism. These two forces are intimately connected and are explained through the theory of electromagnetism. The interplay between electricity and magnetism is visibly demonstrated in scenarios involving current-carrying wires and magnetic fields. A moving electric charge, like current in a wire, generates its own magnetic field. This field interacts with external magnetic fields, resulting in observable physical forces. This fundamental principle is the cornerstone of technologies like motors and generators. In layman's terms, if you have an electric current close to a magnet, you’ve got both an electric and a magnetic aspect forming a single force we call electromagnetism.

Magnetic Field Strength

Magnetic field strength, also known as magnetic flux density, refers to the concentration of magnetic field lines in a given area. It is a vector quantity, measured in Tesla (T), and signifies the force that a magnetic field exerts on moving charges or other magnetic materials within its influence.

• A stronger magnetic field means that magnetic field lines are more densely packed, leading to a greater force on current-carrying wires.
• In the problem at hand, a magnetic field strength of 0.550 T is crucial since it determines the force applied to the wire.

Understanding magnetic field strength is essential, especially when calculating the force exerted on current-carrying wires within a magnetic field. To grasp this fully, picture magnetic field lines as invisible lines of force around magnets: the closer these lines are, the stronger the field is in that area.

Current-Carrying Wire

A current-carrying wire is simply a wire through which electrical current is flowing. When placed in a magnetic field, this wire experiences a magnetic force. This force is the result of interaction between the magnetic field generated by the current and the external magnetic field.

• The direction and magnitude of this force are dictated by the well-known right-hand rule and the formula: \( F = I L B \sin \theta \).
• In our example, the wire is affected by a magnetic field when the current passes through, creating a perpendicular orientation to the field (\(\theta = 90^\circ\)).

The force on the wire is directly proportional to the current and the length of the wire within the field. Hence, increasing either the current or the wire length interacting with the field will amplify the force. A practical understanding of these interactions allows for effective design and operation of electrical devices like transformers and inductors.