Problem 37 - University Physics with Modern Physics

Question

A single circular current loop 10.0 cm in diameter carries a 2.00-A current. (a) What is the magnetic field at the center of this loop? (b) Suppose that we now connect 1000 of these loops in series within a 500 cm length to make a solenoid 500 cm long. What is the magnetic field at the center of this solenoid? Is it 1000 times the field at the center of the loop in part (a)? Why or why not?

Step-by-Step Solution

Answer

The magnetic field at the center of the loop is \(5.03 \times 10^{-6}\) T. At the solenoid’s center, it is \(5.03 \times 10^{-4}\) T, 100 times larger, not 1000, due to solenoid geometry and effects.

Step 1: Understanding the Formula for Magnetic Field due to a Circular Loop

The magnetic field at the center of a circular loop of radius \( R \) carrying a current \( I \) is given by the formula: \[ B = \frac{\mu_0 I}{2 R} \]where \( \mu_0 \) is the permeability of free space \( \mu_0 = 4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A} \).

Step 2: Calculate the Radius of the Loop

The diameter of the loop is given as 10.0 cm, which means the radius \( R \) is half of the diameter:\[ R = \frac{10.0 \ \text{cm}}{2} = 5.0 \ \text{cm} = 0.050 \ \text{m} \].

Step 3: Compute the Magnetic Field at the Center of a Single Loop

Substitute the given values into the formula for magnetic field:\[ B = \frac{(4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}) \times 2.00 \ \text{A}}{2 \times 0.050 \ \text{m}} \]\[ B = \frac{(8\pi \times 10^{-7})}{0.10} \ \text{T} \approx 5.03 \times 10^{-6} \ \text{T} \].

Step 4: Understanding the Formula for a Solenoid

The magnetic field inside a long solenoid is given by:\[ B = \mu_0 n I \]where \( n \) is the number of turns per unit length and \( I \) is the current.

Step 5: Calculate the Number of Turns per Unit Length

Given that there are 1000 loops in a solenoid of length 500 cm (5.00 m), \[ n = \frac{1000}{5.00 \ \text{m}} = 200 \ \text{turns/m} \].

Step 6: Compute the Magnetic Field at the Center of the Solenoid

Substitute the values into the solenoid magnetic field formula:\[ B = (4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}) \times 200 \ \text{turns/m} \times 2.00 \ \text{A} \]\[ B = 1600\pi \times 10^{-7} \ \text{T} \approx 5.03 \times 10^{-4} \ \text{T} \].

Step 7: Analyze the Relationship Between the Fields

The magnetic field at the center of the solenoid is \( 5.03 \times 10^{-4} \ \text{T} \), which is 100 times larger than the field due to a single loop (\( 5.03 \times 10^{-6} \ \text{T} \)). This indicates that while each loop contributes to the total field, the structure of the solenoid changes the amplification factor from 1000 to 100 due to field overlap, edge effects, and geometry.

Key Concepts

Circular Current Loop

A circular current loop is a simple yet fascinating concept in electromagnetism. Imagine a loop made of conducting material with a current flowing through it. This arrangement creates a magnetic field around the loop. The strength and direction of this field depend on factors like the radius of the loop and the current flowing through it.
To compute the magnetic field at the center of a circular loop, we use the formula:

• \( B = \frac{\mu_0 I}{2 R} \)

where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( R \) is the radius of the loop.
This formula highlights that increasing the current or decreasing the radius results in a stronger magnetic field. The field lines form concentric circles around the wire, converging strongly at the center, making it the point of maximum field strength.
Understanding this concept helps explain more complex magnetic systems, such as solenoids, by focusing on how individual loops contribute to larger magnetic fields.

Magnetic Field Formula

To understand magnetic fields, one must first grasp the relevant formulas that allow us to calculate the strength and direction of these fields. As mentioned earlier, the magnetic field at the center of a circular current loop is given by:

• \( B = \frac{\mu_0 I}{2 R} \)

This formula is pivotal because it helps us understand the interaction between electricity and magnetism at a basic level. When calculating the magnetic field, we derive it by considering the contributions from every segment of the current-carrying wire.
For solenoids, the formula becomes:

• \( B = \mu_0 n I \)

Here, \( n \) represents the number of turns per unit length. This difference in the formula reflects how the magnetic field is structured within a solenoid compared to a single loop. Although it appears linear and straightforward, the understanding of these two formulas is key in transitioning from basic electromagnetic theory to practical applications like electromagnets.

Solenoid

A solenoid is essentially a coil of wire that functions as an electromagnet when an electric current passes through it. This structure amplifies the magnetic field significantly compared to a single loop of wire.
In solenoids, closely packed loops create a uniform magnetic field inside, aligning in the direction of the coil. The formula for calculating the magnetic field of a solenoid involves both the current and the number of turns per unit length:

• \( B = \mu_0 n I \)

This formula indicates that both the number of loops and the current directly influence the field's strength. Often, solenoids are long enough that the field outside is negligible, concentrating the magnetic effects within.
By arranging loops in this spiral manner, a solenoid efficiently channels magnetic force lines inside, making it useful for various applications, like in transformers, inductors, or MRI machines. Understanding the dynamics of a solenoid provides valuable insight into electromagnetic devices and their functionalities.

Permeability of Free Space

Permeability of Free Space, often denoted as \( \mu_0 \), is a fundamental constant in electromagnetism that characterizes the capability of a vacuum to support magnetic fields. Its standard value is approximately \( 4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A} \). This value is crucial when calculating magnetic fields in both circular loops and solenoids.
In simple terms, \( \mu_0 \) aids in determining how strong a magnetic field will be in a vacuum when a certain amount of current flows through a conductor. It appears in the formulas for magnetic fields, revealing that it acts as a proportionality constant linking magnetic field strength to factors like current and geometry.
Understanding this constant helps comprehend the theoretical limits of magnetic field generation in vacuums. However, it serves as a baseline for comparing how various materials respond magnetically, making it crucial in the study of electromagnetism.