91Ó°ÊÓ

Inductance of a Solenoid. (a) A long, straight solenoid has \(N\) turns, uniform cross-sectional area \(A,\) and length \(1 .\) Show that the inductance of this solenoid is given by the equation \(L=\mu_{0} A N^{2} / l\) Assume that the magnetic field is uniform inside the solenoid and zero outside. (Your answer is approximate because \(B\) is actually smaller at the ends than at the center. For this reason, your answer is actually an upper limit on the inductance.) (b) \(A\) metallic laboratory spring is typically \(5.00 \mathrm{~cm}\) long and \(0.150 \mathrm{~cm}\) in diameter and has 50 coils. If you connect such a spring in an clectric circuit, how much self-inductance must you include for it if you model it as an idcal solcnoid?

Step by step solution

01

Identify Known Variables

In this problem, we know that the number of turns (\(N\)), the cross-sectional area (\(A\)), and the length (\(l\)) of the solenoid. Additionally, the magnetic constant (\(\mu_{0}\)) is given by \(\mu_{0} = 4\pi \times 10^{-7} T\cdot m/A\). The first task is to prove \(L=\mu_{0} A N^{2} / l\).

02

Review Solenoid Magnetic Field

The magnetic field \(B\) inside a long solenoid carrying a current \(I\) can be calculated with the expression \(B = \mu_{0} \cdot N \cdot I / l \), where \(N\) is the total number of turns and \(l\) is the length of the solenoid. This expression comes from Ampère's law.

03

Express Inductance in terms of Magnetic Field

Inductance \(L\) is expressed using the formula \(L = N \cdot \Phi / I\), where \(\Phi\) is the magnetic flux. Magnetic flux \(\Phi\) through the cross-sectional area \(A\) of the solenoid is given by \(\Phi = B \cdot A\). Substitution of \(\Phi\) and \(B\) will provide the required inductance.

04

Substitute and Simplify

Firstly, substitute for \(\Phi\) in the inductance equation \(L = N \cdot B \cdot A / I\). The magnetic field expression \(B = \mu_{0} \cdot N \cdot I / l \) can be substituted into the equation, resulting in \(L = N \cdot (\mu_{0} \cdot N \cdot I / l) \cdot A / I\). Simply this to achieve \(L = \mu_{0} \cdot A \cdot N^{2} / l\).

05

Calculate Solenoid Inductance

Now, use the derived formula and provided values to calculate the inductance of the metallic laboratory spring. Convert all the dimensions into metres. The length \(l\) is 0.05 m, cross-sectional area \(A = \pi (d/2)^{2}\) is \( 0.150 \times 10^{-2} m = 1.766 \times 10^{-5} m^{2}\), and total number of coils \(N\) is 50. Substitute these into the formula to obtain the inductance.

06

Final Calculation

The inductance \(L\) of the solenoid is calculated as follows: \(L = \mu_{0} \cdot A \cdot N^{2} / l = 4\pi \times 10^{-7} T\cdot m/A \cdot 1.766 \times 10^{-5} m^{2} \cdot (50)^{2} / 0.05 m = 0.22 H\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solenoid

A solenoid is a coil of wire that is designed to generate a magnetic field when an electric current passes through it. It typically consists of many turns of wire, each contributing to the overall magnetic field inside the coil. This design helps to concentrate the magnetic field lines in the center of the solenoid, creating a uniform and strong magnetic field along its length.

• Solenoids are often used in electromagnetic applications, acting as electromagnets.
• The strength of the magnetic field depends on the number of turns, the current flowing through the coil, and the length of the solenoid.
• In practical scenarios, the magnetic field outside of the solenoid is considered negligible compared to inside it.

When designing solenoids for specific applications, engineers carefully choose the parameters such as coil turns and dimensions to achieve the desired magnetic properties.

Magnetic Field

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, magnetic materials, and the intrinsic magnetic moments of elementary particles. Inside a solenoid, the magnetic field B is concentrated and can be calculated using \[ B = \mu_{0} \frac{N I}{l} \], where \( \mu_{0} \) is the permeability of free space, \( N \) represents the number of turns, \( I \) is the current in the solenoid, and \( l \) stands for the solenoid's length.

• The formula shows the direct proportionality of the magnetic field with current and number of turns.
• The magnetic field in a solenoid is uniform at the center but tapers off slightly at the ends.
• Outside a solenoid, the magnetic field is minimal due to cancellation of fields.

This makes solenoids ideal for creating controlled magnetic fields in electrical circuits, components, and different scientific experiments.

Ampère's Law

Ampère's Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is mathematically represented as: \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_{0} I \], where the left side is the integral of the magnetic field B along an imaginary loop, and I is the current passing through any surface bounded by the loop.

• Ampère's Law is crucial in the analysis and understanding of solenoid magnetic fields.
• For a long solenoid, Ampère's Law accurately predicts a uniform magnetic field inside the solenoid, with very little magnetic field outside.
• This law helps in deriving the expression for the magnetic field inside solenoids and other symmetrically structured current distributions.

Ampère's Law is foundational in electromagnetism and is an integral part of designing circuits and electrical systems.

Magnetic Flux

Magnetic flux (\( \Phi \)) quantifies the number of magnetic field lines passing through a given cross-sectional area. It is given by \( \Phi = B \cdot A \), where B is the magnetic field, and A is the area through which the field lines pass.

• Magnetic flux is a product of the average magnetic field times the perpendicular area it penetrates.
• In the context of a solenoid, it helps in calculating inductance as it directly relates to how much magnetic field is "captured" by the coil loops.
• A change in magnetic flux in a circuit induces an electromotive force (EMF), as per Faraday's law of induction.

Understanding magnetic flux is vital in studying inductance and explaining how solar and electrical devices work.