Question

A long, straight wire carries a current of 2.5 A. a) What is the strength of the magnetic field at a distance of \(3.9 \mathrm{~cm}\) from the wire? b) If the wire still carries \(2.5 \mathrm{~A}\), but is used to form a long solenoid with 32 turns per centimeter and a radius of \(3.9 \mathrm{~cm}\), what is the strength of the magnetic field inside the solenoid?

Short answer

Question: Calculate the strength of the magnetic field at a distance of 3.9 cm from a long, straight wire carrying a current of 2.5 A and the magnetic field strength inside a long solenoid with 32 turns per centimeter and a radius of 3.9 cm if the wire carries the same current. Answer: The strength of the magnetic field at a distance of 3.9 cm from the wire is approximately \(6.41 \times 10^{-5}\mathrm{~T}\), and the strength of the magnetic field inside the solenoid is approximately 0.080 T.

Step-by-step solution

Applying Ampere's Law

Using Ampere's Law, the magnetic field around a long, straight wire can be calculated using the formula: \[B = \frac{\mu_0 I}{2 \pi r}\] where: - \(B\) is the magnetic field strength (in T) - \(\mu_0\) is the permeability of free space \((4 \pi \times10^{-7}~ \mathrm{T \cdot m/A})\) - \(I\) is the current in the wire (in A) - \(r\) is the distance from the wire (in m)

Insert the given values into the formula

The current in the wire is 2.5 A and the distance from the wire is 3.9 cm. First, we need to convert the distance from cm to m: \[3.9 \mathrm{~cm} = 0.039 \mathrm{~m}\] Now, plug the values into the formula: \[B = \frac{4 \pi \times10^{-7}~\mathrm{T \cdot m/A} \cdot 2.5~\mathrm{A}}{2 \pi \cdot 0.039~\mathrm{m}}\]

Calculate the magnetic field strength

Simplify the equation to find the magnetic field strength: \[B = \frac{2.5 \cdot 4 \pi \times10^{-7}}{2 \pi \cdot 0.039}\] \[B = \frac{2.5 \cdot 4 \times10^{-7}}{0.039}\] \[B \approx 6.41 \times 10^{-5} \mathrm{~T}\] So the strength of the magnetic field at a distance of 3.9 cm from the wire is approximately \(6.41 \times 10^{-5}\mathrm{~T}\). #b) Calculating Magnetic Field Strength inside the Solenoid#

Applying the formula for Magnetic Field inside a solenoid

The magnetic field strength inside a solenoid can be calculated using the formula: \[B = \mu_0 n I\] where: - \(B\) is the magnetic field strength (in T) - \(\mu_0\) is the permeability of free space \((4 \pi \times10^{-7}~ \mathrm{T \cdot m/A})\) - \(n\) is the number of turns per unit length (in turns/m) - \(I\) is the current in the wire (in A)

Calculate the number of turns per unit length

The given number of turns per centimeter is 32 turns/cm. We need to convert it to turns/m: \[32 \,\text{turns/cm} = 3200 \,\text{turns/m}\]

Insert the given values into the formula for Magnetic Field

Plug the values into the formula: \[B = 4 \pi \times10^{-7}~\mathrm{T \cdot m/A} \cdot 3200~\mathrm{turns/m} \cdot 2.5~\mathrm{A}\]

Calculate the magnetic field strength inside the solenoid

Simplify the equation to find the magnetic field strength inside the solenoid: \[B = 4 \pi \times10^{-7} \cdot 3200 \cdot 2.5\] \[B \approx 0.080~\mathrm{T}\] So, the strength of the magnetic field inside the solenoid is approximately 0.080 T.

Key concepts

Ampere's Law

Understanding Ampere's Law is crucial for students grappling with electromagnetism. This fundamental law relates to the magnetic field around a conductor that carries electric current. Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability of free space times the electric current enclosed in the loop.

In simpler terms, it tells us that the magnetic field circling a current has a certain strength, proportional to the electric current and inversely proportional to the distance from the current-carrying wire. Mathematically, it is expressed as \[B = \dfrac{\mu_0 I}{2 \pi r}\] where \(B\) is the magnetic field, \(I\) is the current, \(r\) is the radius or distance from the wire, and \(\mu_0\) is the permeability of free space. This equation allows the calculation of the magnetic field strength at any point at a known distance from a long straight wire, providing a solid understanding of the relationship between electricity and magnetism.

Magnetic Field Inside a Solenoid

Diving into the magnetic fields in solenoids, understanding their nature and calculation is fascinating. A solenoid is a type of electromagnet whose purpose is to generate a controlled magnetic field. If the solenoid is long compared to its diameter, we can assume the magnetic field inside is uniform and parallel to the axis of the solenoid.

The way to calculate this field is quite elegant and depends on a few factors, especially the number of turns of wire and the current flowing through it. The formula given by \[B = \mu_0 n I\] is deceptively straightforward, encapsulating the relationship between magnetic field strength \(B\), the permeability of free space \(\mu_0\), the number of turns per unit length \(n\), and the current \(I\). Here, \(n\) is the number of turns per length, which means you need more turns for a stronger magnetic field. This direct correlation allows a straightforward calculation by inserting the number of turns per length and the current into the formula.

Permeability of Free Space

Permeability of free space, represented by \(\mu_0\), is a constant that forms the proportionality between magnetic fields and electric current in vacuum. Its value is exactly \(4\pi \times 10^{-7}\) Tesla meter per ampere (T⋅m/A), as established by the International System of Units (SI).

It's often referred to as the magnetic constant, symbolizing how much resistance is encountered when forming a magnetic field in a classical vacuum. Positively, \(\mu_0\) factors into calculations like those for a magnetic field created by current in a wire or magnetic field inside a solenoid. Understanding this constant is fundamental—not only does it appear in Ampere's Law but it also is the basis for many other magnetism-related laws and equations.

In the context of solenoids or wires carrying currents, \(\mu_0\) serves as a bridge between the electrical current and the resulting magnetic field. This sheds light on the intrinsic link between electricity and magnetism, two of the core pillars in the study of physical science.