Problem 40 - University Physics with Modern Physics

Question

A 15.0 -cm-long solenoid with radius 2.50 \(\mathrm{cm}\) is closely wound with 600 turns of wire. The current in the windings is 8.00 A. Compute the magnetic field at a point near the center of the solenoid.

Step-by-Step Solution

Answer

The magnetic field near the center of the solenoid is approximately 0.0402 T.

Step 1: Understand the Formula for Magnetic Field in a Solenoid

The magnetic field inside a long solenoid can be calculated using the formula \( B = \mu_0 \cdot n \cdot I \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \; \text{T}\cdot\text{m/A} \)), \( n \) is the number of turns per unit length, and \( I \) is the current.

Step 2: Calculate the Number of Turns per Unit Length

Calculate \( n \), the number of turns per unit length, using \( n = \frac{N}{L} \), where \( N = 600 \) is the total number of turns, and \( L = 15.0 \; \text{cm} = 0.150 \; \text{m} \) is the length of the solenoid. Thus, \( n = \frac{600}{0.150} = 4000 \; \text{turns/m} \).

Step 3: Substitute the Values into the Magnetic Field Formula

Substitute \( n = 4000 \; \text{turns/m} \) and \( I = 8.00 \; \text{A} \) into the formula \( B = \mu_0 \cdot n \cdot I \). Calculate \( B = (4\pi \times 10^{-7}) \cdot 4000 \cdot 8.00 \).

Step 4: Calculate the Magnetic Field

Carry out the calculation: \( B = 4\pi \times 10^{-7} \cdot 4000 \cdot 8.00 = 4.02 \times 10^{-2} \; \text{T} \). Thus, the magnetic field at a point near the center of the solenoid is approximately \( 0.0402 \; \text{T} \).

Key Concepts

Solenoid

A solenoid is a long coil of wire wound in a helix shape, often used to generate uniform magnetic fields or create electromagnets.
When a current flows through this coil, it generates a magnetic field inside and around the solenoid. This field is typically strongest and most uniform inside the solenoid. The strength of the magnetic field in a solenoid is influenced by several factors:

• Number of Turns: Increasing the number of turns in the wire coil can enhance the magnetic field strength.
• Current: A higher current passing through the wire will result in a stronger magnetic field.
• Length of the Solenoid: The length of the solenoid determines the field's distribution; a longer solenoid will have a more uniform field.

In our exercise, the solenoid has 600 turns and a specific length and radius, all contributing to determining the resulting magnetic field strength when a current of 8.00 A is applied. Understanding the structure and purpose of a solenoid is crucial in calculating its magnetic field.

Electromagnetism

Electromagnetism is the branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. It is a fundamental interaction in nature and is responsible for practically all phenomena encountered in daily life, aside from gravity.
When current flows through a conductor, like the wire in a solenoid, it produces a magnetic field, an essential principle in electromagnetism.

• Electric Current: The flow of electric charge, generally in the form of electrons flowing through a wire, generating a magnetic field.
• Magnetic Fields: Created by moving charges (currents), such as those in a solenoid, these fields can exert forces on moving charges and other magnetic fields.
• Interaction: The interaction between electric currents and magnetic fields is exploited in various technologies, such as electric motors and transformers.

In our case, electromagnetism explains why a magnetic field is induced in the solenoid when an electric current is applied through its windings.

Permeability of Free Space

The permeability of free space, denoted as \( \mu_0 \), is a physical constant essential in electromagnetism. It describes how well a material (or vacuum) supports the formation of a magnetic field. In mathematical terms, it relates the magnetic field strength to the magnetic induction.
The value of the permeability of free space is \( 4\pi \times 10^{-7} \; \text{T}\cdot\text{m/A} \). It appears in various fundamental equations of electromagnetism, such as the formula for calculating the magnetic field inside a solenoid: \( B = \mu_0 \cdot n \cdot I \).

• Magnetic Field: The magnetic field inside a solenoid is directly proportional to the permeability of free space, the number of turns per unit length, and the current flowing through the solenoid.
• Vacuum: While \( \mu_0 \) applies to a vacuum, it also serves as a reference point for other materials with different magnetic properties.

Understanding the role of \( \mu_0 \) is vital for predicting and calculating the strength of magnetic fields in a variety of contexts, particularly in our solenoid calculation exercise.