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Ch 30: Inductance
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 30: InductanceProblem 19c
Chapter 30, Problem 19c

A solenoid 25.0 cm long and with a cross-sectional area of 0.500 cm2 contains 400 turns of wire and carries a current of 80.0 A. Calculate: the total energy contained in the coil's magnetic field (assume the field is uniform);

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First, understand that the energy stored in the magnetic field of a solenoid can be calculated using the formula for the energy density of a magnetic field: \( u = \frac{1}{2} \frac{B^2}{\mu_0} \), where \( B \) is the magnetic field and \( \mu_0 \) is the permeability of free space.
Calculate the magnetic field \( B \) inside the solenoid using the formula \( B = \mu_0 n I \), where \( n \) is the number of turns per unit length (\( n = \frac{N}{L} \)), \( N \) is the total number of turns, \( L \) is the length of the solenoid, and \( I \) is the current.
Substitute the values into the formula for \( n \): \( n = \frac{400}{0.25} \) turns/m. Then calculate \( B \) using \( B = \mu_0 n I \).
Once \( B \) is calculated, find the energy density \( u \) using \( u = \frac{1}{2} \frac{B^2}{\mu_0} \).
Finally, calculate the total energy stored in the solenoid by multiplying the energy density \( u \) by the volume of the solenoid: \( E = u \times \text{Volume} \), where the volume is \( A \times L \) and \( A \) is the cross-sectional area.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Field in a Solenoid

The magnetic field inside a long solenoid is uniform and can be calculated using the formula B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. This concept is crucial for understanding how the solenoid generates a magnetic field.
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Magnetic Energy Density

Magnetic energy density is the energy stored per unit volume in a magnetic field, given by the formula u = B²/(2μ₀), where u is the energy density, B is the magnetic field, and μ₀ is the permeability of free space. This concept helps in determining how much energy is stored in the magnetic field of the solenoid.
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Total Magnetic Energy in a Solenoid

The total energy stored in the magnetic field of a solenoid can be calculated by multiplying the magnetic energy density by the volume of the solenoid. The volume is found by multiplying the cross-sectional area by the length of the solenoid. This concept is essential for calculating the total energy contained in the solenoid's magnetic field.
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Related Practice
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A long, straight solenoid has 800 turns. When the current in the solenoid is 2.90 A, the average flux through each turn of the solenoid is 3.25 × 10-3 Wb. What must be the magnitude of the rate of change of the current in order for the self-induced emf to equal 6.20 mV?

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Textbook Question

Inductance of a Solenoid. A metallic laboratory spring is typically 5.00 cm long and 0.150 cm in diameter and has 50 coils. If you connect such a spring in an electric circuit, how much self-inductance must you include for it if you model it as an ideal solenoid?

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Textbook Question

It has been proposed to use large inductors as energy storage devices. If the amount of energy calculated in part is stored in an inductor in which the current is 80.0 A, what is the inductance?

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