Problem 67 - College Physics

Question

\bullet A single circular current loop 10.0 \(\mathrm{cm}\) 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 \(\mathrm{cm}\) length to make a solenoid 500 \(\mathrm{cm}\) 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

(a) The field is approximately \(1.257 \times 10^{-5} \, \text{T}\). (b) The solenoid's field is approximately \(0.0002513 \, \text{T}\), about 20 times the single loop's field, not 1000 times.

Step 1: Calculate Radius of the Loop

First, determine the radius of the single loop. The diameter is given as \(10.0 \, \text{cm}\), so the radius is half of that. The radius \(r\) is \(\frac{10.0}{2} = 5.0 \, \text{cm} = 0.05 \, \text{m}\).

Step 2: Calculate Magnetic Field at Center of Single Loop

Use the formula for the magnetic field \(B\) at the center of a single current-carrying loop: \[ B = \frac{\mu_0 I}{2r} \]where \( \mu_0 \) is the permeability of free space \( (4 \pi \times 10^{-7} \, \text{T}\cdot\text{m/A}) \) and \( I = 2.00 \, \text{A} \). Substitute the values:\[B = \frac{4 \pi \times 10^{-7} \times 2.00}{2 \times 0.05} = \frac{8 \pi \times 10^{-7}}{0.1} = \frac{8 \pi \times 10^{-7}}{0.1} \, \text{T} \approx 1.257 \times 10^{-5} \, \text{T}\]

Step 3: Analyze Solenoid Configuration

With 1000 loops in series, the solenoid acts like a long coil. Use the formula for the magnetic field at the center of a long solenoid:\[ B = \mu_0 \frac{N}{L} I \]where \( N = 1000 \) loops and \( L = 5.00 \, \text{m} \). Substitute the values:\[B = 4 \pi \times 10^{-7} \times \frac{1000}{5.00} \times 2.00\]\[B = 4 \pi \times 10^{-7} \times 200 = 800 \pi \times 10^{-7} \, \text{T} \approx 0.0002513 \, \text{T}\]

Step 4: Compare Solenoid Field with Single Loop Field

The solenoid's magnetic field is calculated as approximately \(0.0002513 \, \text{T}\), while the magnetic field of a single loop was \(1.257 \times 10^{-5} \, \text{T}\). To find the factor of increase:\[\text{Factor} = \frac{0.0002513}{1.257 \times 10^{-5}} \approx 20\]The magnetic field of the solenoid is not 1000 times the field of a single loop, it is about 20 times larger, as the formula depends on the geometry and distribution of the loops, not merely the number of loops.

Key Concepts

Circular Current Loop

A circular current loop is a fundamental concept in electromagnetism. It consists of a loop of wire carrying an electric current, which creates a magnetic field around itself. The magnetic field is strongest at the center of the loop and decreases as you move towards the edges.

The magnetic field at the center of a circular current loop can be calculated using the formula:

• \[ B = \frac{\mu_0 I}{2r} \]

Where:

• \( B \) is the magnetic field measured in teslas (T),
• \( \mu_0 \) is the permeability of free space,
• \( I \) is the current through the loop in amperes (A),
• \( r \) is the radius of the loop in meters.

This configuration is fundamental to understanding how loops and turns of wire can generate magnetic fields and is a building block for more complex systems like solenoids.

Solenoid

A solenoid is essentially a coil of wire, often wrapped in the shape of a cylinder, through which electric current flows. When current passes through, it creates a magnetic field similar to a bar magnet, with distinct north and south poles. The strength and direction of the magnetic field in a solenoid can be influenced by factors such as coil length, the number of loops, and the current flowing through it.

The magnetic field inside a long solenoid is uniform and can be calculated using:

• \[ B = \mu_0 \frac{N}{L} I \]

Where:

• \( B \) is the magnetic field in teslas (T),
• \( \mu_0 \) is the permeability of free space,
• \( N \) is the number of turns or loops of the coil,
• \( L \) is the length of the solenoid in meters,
• \( I \) is the current in amperes.

The field outside the solenoid is practically negligible, making this device very useful in applications that require a controllable and localized magnetic field such as electromagnets, transformers, and inductors.

Permeability of Free Space

The permeability of free space, often noted as \( \mu_0 \), is a fundamental physical constant important in electromagnetism. It represents the extent to which a magnetic field can penetrate the vacuum of space. This constant is crucial in determining the strength of a magnetic field created by an electric current.

The value of \( \mu_0 \) is approximately equal to \( 4 \pi \times 10^{-7} \) Tesla meter per Ampere (T·m/A). This value is used universally in calculations involving magnetic fields and current-carrying conductors.

• It's essential in equations defining magnetic forces, the magnetic field's strength, and the interaction between electric currents and magnetic fields.

Understanding \( \mu_0 \) helps us predict how magnetic fields behave in different environments, particularly in a vacuum or air, which simplifies to this universal permeability.

Magnetic Field Formula

The magnetic field formulae are central to physics, allowing us to quantify the magnetic fields generated by currents. Two key formulae are commonly employed: one for a single loop and another for a solenoid.

For a circular current loop:

• \[ B = \frac{\mu_0 I}{2r} \]

This formula estimates the magnetic field at the center of a loop and depends on how much current flows and the loop's size.

For a solenoid:

• \[ B = \mu_0 \frac{N}{L} I \]

This formula considers the solenoid's length, the number of turns, and the current to determine the field inside it. Both highlight the role of \( \mu_0 \) in expressing the magnetic properties of different geometries and their strength based on current and physical characteristics.

• They allow comparisons between different systems, illustrating how the field's intensity changes based on design and materials used.