Problem 65 - College Physics

Question

\(\bullet\) As a new electrical technician, you are designing a large solenoid to produce a uniform 0.150 T magnetic field near its center. You have enough wire for 4000 circular turns, and the solenoid must be 1.40 m long and 2.00 \(\mathrm{cm}\) in diameter. What current will you need to produce the necessary field?

Step-by-Step Solution

Answer

The current needed is approximately 41.8 A.

Step 1: Understand the Formula for Solenoid's Magnetic Field

The magnetic field inside a solenoid is given by 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 Turns per Unit Length

The number of turns per unit length \(n\) is calculated as the total number of turns divided by the length of the solenoid. Here, \(n = \frac{4000}{1.40} \, \text{turns/m}\).

Step 3: Rearrange the Formula to Solve for Current

Rearrange the formula to find the current: \(I = \frac{B}{\mu_0 \cdot n}\). This allows us to solve for \(I\) once \(n\) is known.

Step 4: Substitute Values and Solve for Current

Substitute the known values into the rearranged formula: \(I = \frac{0.150 \, \text{T}}{4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A} \cdot \frac{4000}{1.40} \, \text{turns/m}}\). Calculate \(I\) to find the current required.

Step 5: Calculate the Numerical Answer

Calculate \(I\) using the substituted values:\[I = \frac{0.150}{4\pi \times 10^{-7} \times \frac{4000}{1.40}}\]This results in \(I \approx 41.8 \, \text{A}\).

Key Concepts

Magnetic Field

A magnetic field is a fundamental concept in physics, relating to a region where a magnetic force can be detected. In the context of solenoids, a magnetic field is created when an electric current flows through the wire loops. This field is uniform in the center of a solenoid, which means it has a constant strength and direction at that point. The strength of the magnetic field inside a solenoid is given by the formula \( B = \mu_0 \cdot n \cdot I \), 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 solenoid.

• Uniform fields are ideal for applications like electromagnets and transformers, where consistent magnetic effects are required.
• The strength of the magnetic field is directly proportional to current and the number of turns per unit length.

Current Calculation

Calculating the current needed to create a desired magnetic field inside a solenoid is crucial. The steps involve using the formula \( I = \frac{B}{\mu_0 \cdot n} \). This requires re-arranging the fundamental solenoid magnetic field equation.
First, understand that the current \( I \) is what powers the magnetic field. To find \( I \), you need to know the desired magnetic field \( B \), the permeability of free space \( \mu_0 \), and the number of turns per unit length \( n \). Substituting these values into the formula gives you the current required.
Consider this example: if you need \( 0.150 \) Tesla magnetic field from a solenoid, with permeability \( 4\pi \times 10^{-7} \) T\cdot m/A, and \( n = \frac{4000}{1.40} \) turns/m, you find the current needed is approximately \( 41.8 \) A.

Turns per Unit Length

The term "turns per unit length" refers to how many loops of wire are packed into each meter of the solenoid's length. It is symbolized by \( n \) in equations. This value significantly impacts the strength of the magnetic field produced by the solenoid. The calculation is straightforward: divide the total number of turns by the solenoid's length. In our example, the solenoid has 4000 turns and a length of 1.40 meters. Therefore, \( n = \frac{4000}{1.40} \) turns/m.

• Higher number of turns per unit length increases the magnetic field strength.
• Compact solenoids can save space while still producing a strong magnetic field.

This concept helps in designing solenoids with desired field strengths by adjusting the number of turns and their distribution along the solenoid's length.

Permeability of Free Space

The permeability of free space, symbolized by \( \mu_0 \), is a constant that appears in several magnetic equations. It quantifies how easily a magnetic field can be established in a vacuum. The value of \( \mu_0 \) is approximately \( 4\pi \times 10^{-7} \) T\cdot m/A. This constant is critical when calculating the magnetic field inside a solenoid. When combined with the current and turns per unit length, it determines the field’s strength.

• Acts as a benchmark to compare with materials having different permeability.
• The uniformity of this constant simplifies calculations in theoretical physics.

Understanding \( \mu_0 \) helps predict how materials will respond to magnetic fields, crucial in designing devices like transformers, inductors, and other electromagnetic applications.