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Ch 30: Inductance
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 30: InductanceProblema 17
Capitolo 30, Problema 17

An air-filled toroidal solenoid has a mean radius of 15.0 cm and a cross-sectional area of 5.00 cm2. When the current is 12.0 A, the energy stored is 0.390 J. How many turns does the winding have?

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Start by recalling the formula for the energy stored in the magnetic field of a solenoid: \( U = \frac{1}{2} L I^2 \), where \( U \) is the energy stored, \( L \) is the inductance, and \( I \) is the current.
Rearrange the formula to solve for the inductance \( L \): \( L = \frac{2U}{I^2} \). Substitute \( U = 0.390 \ \text{J} \) and \( I = 12.0 \ \text{A} \) into the equation to calculate \( L \).
Next, use the formula for the inductance of a toroidal solenoid: \( L = \mu_0 \frac{N^2 A}{2 \pi r} \), where \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \ \text{T·m/A} \)), \( N \) is the number of turns, \( A \) is the cross-sectional area, and \( r \) is the mean radius.
Rearrange the formula to solve for \( N \): \( N = \sqrt{\frac{L \cdot 2 \pi r}{\mu_0 A}} \). Substitute \( L \) from step 2, \( r = 15.0 \ \text{cm} = 0.150 \ \text{m} \), and \( A = 5.00 \ \text{cm}^2 = 5.00 \times 10^{-4} \ \text{m}^2 \) into the equation.
Simplify the expression to calculate \( N \), the number of turns in the winding. Ensure all units are consistent (meters, henries, etc.) during the calculation.

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Toroidal Solenoid

A toroidal solenoid is a coil of wire shaped like a doughnut, where the magnetic field is confined within the loop. The magnetic field strength inside a toroidal solenoid depends on the number of turns per unit length and the current flowing through the wire. This configuration allows for a uniform magnetic field and is often used in applications requiring compact magnetic fields.
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Toroidal Solenoids aka Toroids

Magnetic Energy Storage

The energy stored in a magnetic field can be calculated using the formula U = (1/2)LI², where U is the energy, L is the inductance, and I is the current. In a toroidal solenoid, the inductance depends on the geometry of the solenoid, including the number of turns, the mean radius, and the cross-sectional area. Understanding this relationship is crucial for determining how much energy is stored based on the current and the solenoid's physical characteristics.
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Magnetic Fields and Magnetic Dipoles

Inductance Calculation

Inductance is a measure of how much magnetic flux is generated for a given current in a coil. For a toroidal solenoid, the inductance can be calculated using the formula L = (μ₀N²A)/2πr, where μ₀ is the permeability of free space, N is the number of turns, A is the cross-sectional area, and r is the mean radius. This formula is essential for solving problems related to the number of turns in a solenoid when other parameters are known.
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Self Inductance
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