Problem 41 - University Physics with Modern Physics

Question

A solenoid is designed to produce a magnetic field of 0.0270 \(\mathrm{T}\) at its center. It has radius 1.40 \(\mathrm{cm}\) and length \(40.0 \mathrm{cm},\) and the wire can carry a maximum current of 12.0 \(\mathrm{A}\) . (a) What minimum number of turns per unit length must the solenoid have? (b) What total length of wire is required?

Step-by-Step Solution

Answer

(a) Minimum turns per unit length: 1786 turns/m (b) Total wire length: 62.9 m

Step 1: Understand the Magnetic Field Formula for a Solenoid

The formula for the magnetic field at the center of a solenoid is given by \( B = \mu_0 \cdot n \cdot I \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length, and \( I \) is the current flowing through the wire. For this exercise, we are given \( B = 0.0270 \ \mathrm{T} \), \( I = 12.0 \ \mathrm{A} \), and we need to solve for \( n \).

Step 2: Rearrange and Solve for Turns per Unit Length \( n \)

Use the formula \( B = \mu_0 \cdot n \cdot I \) to solve for \( n \). Rearranging the formula gives us \( n = \frac{B}{\mu_0 \cdot I} \). Substituting the values, \( B = 0.0270 \ \mathrm{T} \), \( \mu_0 = 4\pi \times 10^{-7} \ \mathrm{T \cdot m/A} \), and \( I = 12.0 \ \mathrm{A} \), we find \( n = \frac{0.0270}{4\pi \times 10^{-7} \cdot 12.0} \approx 1786 \ \mathrm{turns/m} \).

Step 3: Calculate Total Length of Wire Required

The total length of wire is determined by multiplying the total number of turns by the length of the solenoid. The total number of turns is equal to \( n \) multiplied by the solenoid's length \( L \). Here, \( L = 0.400 \ \mathrm{m} \), so the total number of turns \( N = n \cdot L = 1786 \ \mathrm{turns/m} \times 0.400 \ \mathrm{m} = 714 \ \mathrm{turns} \). The length of wire needed is approximately the circumference of the solenoid \( 2\pi r \) times the number of turns, where \( r = 0.0140 \ \mathrm{m} \). Thus, the total length is \( N \cdot 2\pi r = 714 \times 2\pi \times 0.0140 \approx 62.9 \ \mathrm{m} \).

Key Concepts

Solenoid

A solenoid is a coil of wire that is designed to generate a magnetic field when an electric current passes through it. It is typically shaped like a cylinder and is often used in devices that require controlled magnetic fields.

Solenoids are widely used in various applications including electromagnets, inductors, transformers, and many kinds of electrical circuits. The main purpose of a solenoid is to convert electrical energy into mechanical motion. When looking at solenoids, it's important to understand the elements that influence the strength of the magnetic field they produce:

• **Current ( I ):** The amount of electric current flowing through the wire. It is directly proportional to the strength of the magnetic field.
• **Turns of Wire:** The number of loops or coils in the solenoid. Increasing the number of turns can amplify the magnetic field.
• **Length of the Solenoid:** Affects the uniformity of the magnetic field; longer solenoids have more uniform fields near the center.

These elements combined define how a solenoid works in a practical setting, making it a versatile component in engineering and technology.

Turns per Unit Length

The term 'turns per unit length' ( n ) is crucial when dealing with solenoids, as it indicates how tightly the coil is wound. It represents the number of loops of wire within a given length of the solenoid. Here's why this is important:

To produce a specific magnetic field strength, accurate calculation of turns per unit length is vital. The magnetic field ( B ) inside an ideal solenoid is calculated using the formula:
\[ B = \mu_0 \cdot n \cdot I \]
Where:

• B is the magnetic field strength.
• \mu_0 represents the permeability of free space (approximately 4\pi \times 10^{-7} T·m/A).
• I indicates the current through the solenoid.

By rearranging this formula, we can determine the necessary turns per unit length for a desired magnetic field: \[ n = \frac{B}{\mu_0 \cdot I} \]
This equation highlights that to maintain a stronger magnetic field in a longer solenoid, you'll need either more current or a higher density of turns of wire. This balance is key in designing effective solenoids for various technological uses.

Wire Length Calculation

When considering the construction of a solenoid, calculating the total length of wire needed is an essential step. This ensures that the solenoid can be built with the materials available, maintaining both efficient production and cost control.

The process involves determining the total number of turns and then calculating the wire length required for those turns. For example:

• **Determine Total Turns:** Calculate the total number of turns by multiplying the turns per unit length ( n ) by the solenoid's length ( L ).
• **Calculate Wire Length:** Multiply the total number of turns by the circumference of the solenoid ( 2\pi r ), where r is the solenoid's radius.

For a solenoid of length 0.400 m with 714 total turns, using a radius of 0.0140 m, the formula gives us:
\[ \text{Wire Length} = N \cdot 2 \pi r = 714 \times 2\pi \times 0.0140 \approx 62.9 \, \text{m} \]
Accurately calculating this helps avoid wasting materials and ensures the solenoid works as intended once complete.