Problem 64 - College Physics

Question

\(\cdot\) A solenoid contains 750 coils of very thin wire evenly wrapped over a length of 15.0 \(\mathrm{cm} .\) Each coil is 0.800 \(\mathrm{cm}\) in diameter. If this solenoid carries a current of \(7.00 \mathrm{A},\) what is the magnetic field at its center?

Step-by-Step Solution

Answer

The magnetic field at the solenoid's center is approximately 0.0440 T.

Step 1: Identify the formula for the magnetic field inside a solenoid

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

Step 2: Calculate the number of turns per unit length

The total number of coils, or turns, is given as 750, and the length of the solenoid is 15.0 cm, which can be converted to meters as 0.15 m. Thus, the number of turns per unit length \( n \) is calculated as \( n = \frac{750}{0.15} \).

Step 3: Plug in the values for the calculation

Substitute the known values into the formula: \( B = (4\pi \times 10^{-7} \ \text{T m/A}) \times (\frac{750}{0.15}) \times 7 \).

Step 4: Calculate the magnetic field

Calculate the result using the values from the previous step, resulting in \( B \approx 0.0440 \ \text{T} \).

Key Concepts

Solenoid Properties

A solenoid is a coil of wire that is often shaped like a cylinder. These coils can create a magnetic field when an electric current passes through them, a principle that is fundamental in many electromagnetic devices. Some key properties of a solenoid to consider include:

• Number of Turns: Each wrap of wire around the cylinder is a turn or coil. The number of turns is directly proportional to the strength of the magnetic field generated.
• Length of Solenoid: This is the total distance over which the wire is wrapped. It's important to note the length impacts how concentrated the turns are, which in turn affects the magnetic field intensity.
• Coil Diameter: Though not directly required for calculating the magnetic field, it plays a role in applications where the spacing of the coils and the physical design are matters of concern.
• Core Type: The core material (often air or iron) inside affects how the magnetic field is distributed and its overall strength. However, for this scenario, we assume an air core.

Solenoids are widely used in electrical and engineering applications due to these properties, especially where precise control of the magnetic field is required.

Magnetic Field Calculation

To find the magnetic field in the center of a solenoid, the formula used is:\[ B = \mu_0 n I \]Where:

• \( B \) is the magnetic field strength.
• \( \mu_0 \) is the permeability of free space, a constant value.
• \( n \), the number of turns per unit length, is calculated by dividing the total number of turns by the solenoid's length.
• \( I \) is the current passing through the solenoid.

Let's break this down further: The number of turns per unit length, \( n \), is calculated by dividing the total number of coils, 750, by the length of the solenoid in meters, 0.15 m. Once you have \( n \), substitute all values into the formula to solve for \( B \), the magnetic field. The calculation uses the known constants and given variables to output the magnetic field in Tesla (T). This straightforward approach makes solenoids a valuable tool for generating controlled magnetic fields in labs and industrial applications.

Permeability of Free Space

The permeability of free space \( \mu_0 \) is a fundamental constant crucial to understanding electromagnetism. Its value is approximately \( 4\pi \times 10^{-7} \ \text{T m/A} \), and it signifies how much resistance is encountered when forming a magnetic field in a vacuum. This concept helps define the relationship between magnetic fields and currents in spaces that are not filled with material that may otherwise amplify or diminish the magnetic effects.

The idea becomes significant when calculating the magnetic field in air-core solenoids like the one discussed. The permeability of free space sets the baseline for determining the field's intensity inside a solenoid, as it works as a proportional constant in the main magnetic field equation. By knowing \( \mu_0 \), we can better predict and manipulate magnetic fields in various devices, from simple electromagnets to advanced equipment, allowing for a broad range of applications in technology and physics.