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Chapter 7: Problem 11

L for a solenoid \(*\) Find the self-inductance of a long solenoid with radius \(r\), length \(\ell\), and \(N\) turns.

Short Answer

Expert verified
The self-inductance, \(L\) of the solenoid, can be found by inserting the given values into the formula \(L = \mu_0 \cdot \frac{N^2}{\ell} \cdot \pi r^2\), then performing the calculation and lastly simplifying the expression.

Step by step solution

01

Insert the Values

Let's firstly insert all the given values like the number of turns \(N\), the radius \(r\), the length \(\ell\), and the permeability of free space \(\mu_0 = 4 \pi \times 10^{-7} Tm/A\) into the formula \(L = \mu_0 \cdot \frac{N^2}{\ell} \cdot \pi r^2\).
02

Perform the Calculation

The values inserted should be processed mathematically to calculate the self-inductance. Some of the operations include squaring the number of turns and the radius, calculating the product of the permeability of free space, and division by length.
03

Simplify the Expression

Once the calculations are done, simplify the expression to its lowest terms if possible to obtain the final answer, that is the self-inductance \(L\) in Henry (H).

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Key Concepts

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

Understanding a Solenoid
A solenoid is a long, cylindrical coil of wire wound in a helical pattern. It is commonly used in electronic circuits to generate a magnetic field or induce an electromagnetic force.

When current flows through the wire, it creates a uniform magnetic field along the axis of the solenoid. The strength of this magnetic field is influenced by several factors:
  • The number of turns of the wire, denoted by \(N\)
  • The current flowing through the wire
  • The physical dimensions like the radius \(r\) and the length \(\ell\) of the solenoid
The self-inductance of a solenoid is a measure of its ability to induce an electromotive force in itself as the current changes. It is crucial in applications like transformers, electromagnets, and inductors.
Permeability of Free Space
The permeability of free space, denoted as \(\mu_0\), is a fundamental constant in physics, symbolizing the ability of a vacuum to support a magnetic field. Its value is \(4 \pi \times 10^{-7} \ \text{Tm/A}\).

This constant plays a vital role in determining the self-inductance of solenoids and other similar devices.
  • It appears in the formula for calculating self-inductance, emphasizing its influence on the strength of the magnetic field generated.
  • Its inclusion simplifies calculations and provides a standard reference point for various magnetic phenomena.
Understanding \(\mu_0\) helps appreciate how different materials react to magnetic fields and aids in designing circuits with predictable behaviors.
Mathematical Calculation for Self-Inductance
To calculate the self-inductance \(L\) of a solenoid, a specific formula is used:
\[L = \mu_0 \cdot \frac{N^2}{\ell} \cdot \pi r^2\]This equation involves several steps:
  • Plug in the number of turns \(N\), radius \(r\), length \(\ell\), and the permeability of free space \(\mu_0\).
  • Square the number of turns \(N\) and the radius \(r\).
  • Multiply these with \(\mu_0\) and \(\pi \).
  • Finally, divide by the length \(\ell\) to find \(L\).
Performing each of these operations leads to the self-inductance measured in Henry \(H\). Simplifying the expression after calculation ensures clarity and accuracy in the final result.

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Most popular questions from this chapter

Maximum emffor a thin loop *** A long straight stationary wire is parallel to the \(y\) axis and passes through the point \(z=h\) on the \(z\) axis. A current \(I\) flows in this wire, returning by a remote conductor whose field we may neglect. Lying in the \(x y\) plane is a thin rectangular loop with two of its sides, of length \(\ell\), parallel to the long wire. The length \(b\) of the other two sides is very small. The loop slides with constant speed \(v\) in the \(\hat{x}\) direction. Find the magnitude of the electromotive force induced in the loop at the moment the center of the loop has position \(x .\) For what values of \(x\) does this emf have a local maximum or minimum? (Work in the approximation where \(b \ll x\), so that you can approximate the relevant difference in \(B\) fields by a derivative.)

Maximum emf * What is the maximum electromotive force induced in a coil of 4000 turns, average radius \(12 \mathrm{~cm}\), rotating at 30 revolutions per second in the earth's magnetic field where the field intensity is \(0.5\) gauss?

Current in a bottle \(*\) An ocean current flows at a speed of 2 knots (approximately \(1 \mathrm{~m} / \mathrm{s}\) ) in a region where the vertical component of the earth's magnetic field is \(0.35\) gauss. The conductivity of seawater in that region is 4 (ohm-m) \(^{-1}\). On the assumption that there is no other horizontal component of \(\mathbf{E}\) than the motional term \(\mathbf{v} \times \mathbf{B}\), find the density \(J\) of the horizontal electric current. If you were to carry a bottle of seawater through the earth's field at this speed, would such a current be flowing in it?

RL circuit ** A coil with resistance of \(0.01\) ohm and self-inductance \(0.50\) millihenry is connected across a large 12 volt battery of negligible internal resistance. How long after the switch is closed will the current reach 90 percent of its final value? At that time, how much energy, in joules, is stored in the magnetic field? How much energy has been withdrawn from the battery up to that time?

L for a cylindrical solenoid ** Calculate the self-inductance of a cylindrical solenoid \(10 \mathrm{~cm}\) in diameter and \(2 \mathrm{~m}\) long. It has a single-layer winding containing a total of 1200 turns. Assume that the magnetic field inside the solenoid is approximately uniform right out to the ends. Estimate roughly the magnitude of the error you will thereby incur. Is the true \(L\) larger or smaller than your approximate result?
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