Problem 45 - University Physics with Modern Physics

Question

A solenoid that is 35 cm long and contains 450 circular coils 2.0 cm in diameter carries a 1.75-A current. (a) What is the magnetic field at the center of the solenoid, 1.0 cm from the coils? (b) Suppose we now stretch out the coils to make a very long wire carrying the same current as before. What is the magnetic field 1.0 cm from the wire's center? Is it the same as that in part (a)? Why or why not?

Step-by-Step Solution

Answer

(a) 0.00283 T; (b) 0.000035 T. The fields are different because a solenoid confines the field while a wire does not.

Step 1: Understanding the Problem

We are given a solenoid with certain dimensions and properties, and we need to calculate the magnetic field at specific points. First, we calculate the magnetic field at the center of the solenoid. Second, we analyze the field when the solenoid is stretched into a straight wire.

Step 2: Formula for Magnetic Field of 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 (\(4\pi \times 10^{-7}\) Tm/A), \( n \) is the number of turns per unit length, and \( I \) is the current through the solenoid.

Step 3: Calculate Number of Turns per Unit Length

The solenoid has 450 turns over a length of 35 cm. Convert the length into meters (0.35 m). Then, the number of turns per unit length \( n = \frac{450}{0.35} \approx 1285.71 \text{ turns/m} \).

Step 4: Calculate Magnetic Field Inside the Solenoid

Using the formula \( B = \mu_0 n I \), substitute \( \mu_0 = 4\pi \times 10^{-7} \), \( n = 1285.71 \text{ turns/m} \), and \( I = 1.75 \text{ A} \). This gives \( B \approx (4\pi \times 10^{-7}) \times 1285.71 \times 1.75 \approx 0.00283 \text{ T} \).

Step 5: Understanding Magnetic Field Near a Long Wire

The magnetic field near a long straight wire is given by \( B = \frac{\mu_0 I}{2\pi r} \), where \( r \) is the distance from the wire. We need to use this formula to calculate the field 1.0 cm away from the wire.

Step 6: Calculate Magnetic Field Near the Wire

Substitute \( \mu_0 = 4\pi \times 10^{-7} \), \( I = 1.75 \text{ A} \), and \( r = 0.01 \text{ m} \) (1.0 cm) into \( B = \frac{\mu_0 I}{2\pi r} \). This results in \( B \approx \frac{(4\pi \times 10^{-7}) \times 1.75}{2\pi \times 0.01} \approx 0.000035 \text{ T} \).

Step 7: Compare Magnetic Fields in Parts (a) and (b)

The magnetic field at the center of the solenoid \(0.00283 \text{ T}\) is much larger than the field near the straight wire \(0.000035 \text{ T}\). This is due to the difference in the configurations; solenoids concentrate magnetic field lines, while a straight wire disperses them.

Key Concepts

Solenoid

A solenoid is a coil of wire that is tightly wound in a helix. This arrangement is commonly used to generate a magnetic field when an electric current is passed through it. A basic characteristic of a solenoid is that its magnetic field is uniform and strong inside the coil compared to the field outside. The magnetic field in a solenoid can be calculated using the formula:

• \( B = \mu_0 n I \)

Where:

• \( B \) is the magnetic field,
• \( \mu_0 \) is the permeability of free space, which is a constant value \(4\pi \times 10^{-7}\) Tm/A,
• \( n \) is the number of turns per unit length of the solenoid,
• \( I \) is the current passing through the wire.

The closer the coils, the stronger the magnetic field, as the term \( n \) represents the density of turns all along the solenoid's length.

Magnetic Field of a Wire

The magnetic field produced by a long, straight wire carrying a current is different than that of a solenoid. Instead of being concentrated, it extends in circular loops around the wire. The strength and direction of the field are given by the right-hand rule and the formula:

• \( B = \frac{\mu_0 I}{2\pi r} \)

Where:

• \( B \) is the magnetic field,
• \( \mu_0 \) is the permeability of free space,
• \( I \) is the current carried by the wire,
• \( r \) is the radial distance from the wire.

This shows that the magnetic field decreases rapidly as you move away from the wire. Unlike solenoids, which concentrate the magnetic field internally, a single wire disperses its field more widely. This difference in field distribution is important when considering the design of electromagnets and other magnetic devices.

Electromagnetism

Electromagnetism is the branch of physics that studies the interaction between electric currents and magnetic fields. Within the realm of electromagnetism, solenoids and wires are crucial components due to their ability to generate magnetic fields. When a current travels through a wire or solenoid, a magnetic field is created, demonstrating the close link between electricity and magnetism. The fundamental principle here is captured by Ampère's Circuital Law, which relates the circulating magnetic field in a closed loop to the electric current passing through the loop. Thus, both solenoids and straight wires illustrate fundamental electromagnetism concepts. These components allow us to control and utilize magnetic fields by adjusting electrical currents, making applications like motors, transformers, and inductors possible.

Current and Magnetism

The close relationship between electric currents and magnetic fields is a cornerstone of physics. When an electric current passes through a wire, it generates a magnetic field perpendicular to the flow of current. This behavior is fundamental in creating electromagnets, which use coils of wire (like solenoids) to produce concentrated magnetic fields when energized with current. In solenoids, the magnetic field inside the coil is significantly stronger than outside because the multiple loops of wire help concentrate the magnetic field lines within. This used in various applications like electrical circuits, actuators, and magnetic resonance imaging machines. In contrast, a single wire will have magnetic fields that are dispersed concentrically around the wire, showing the versatile ways in which electricity can be used to manipulate magnetic effects for varied technological uses.