Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits

Problem 16

Chapter 29, Problem 16

What is the energy density at the center of a circular loop of wire carrying a 19.0-A current if the radius of the loop is 28.0 cm?

Verified step by step guidance

Understand the problem: The energy density at the center of a circular loop of wire is related to the magnetic field generated by the current in the loop. The formula for energy density is $ u = \frac{B^2}{2\mu_0} $, where $ B $ is the magnetic field and $ \mu_0 $ is the permeability of free space ($ \mu_0 = 4\pi \times 10^{-7} \ \text{T·m/A} $).

Calculate the magnetic field $ B $ at the center of the loop using the formula $ B = \frac{\mu_0 I}{2R} $, where $ I $ is the current (19.0 A) and $ R $ is the radius of the loop (28.0 cm or 0.28 m).

Substitute the known values into the formula for $ B $: $ B = \frac{(4\pi \times 10^{-7})(19.0)}{2(0.28)} $. Simplify this expression to find the magnetic field at the center of the loop.

Once $ B $ is determined, substitute it into the energy density formula $ u = \frac{B^2}{2\mu_0} $. Use the value of $ \mu_0 = 4\pi \times 10^{-7} \ \text{T·m/A} $.

Simplify the expression for $ u $ to find the energy density at the center of the loop. Ensure all units are consistent throughout the calculation.

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Key Concepts

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

Magnetic Field Due to a Current Loop

A circular loop of wire carrying an electric current generates a magnetic field around it. The magnetic field strength at the center of the loop can be calculated using the formula B = (μ₀ * I) / (2 * R), where μ₀ is the permeability of free space, I is the current, and R is the radius of the loop. This magnetic field is crucial for understanding the energy density associated with the loop.

Energy Density in a Magnetic Field

The energy density (u) in a magnetic field is defined as the energy stored per unit volume and is given by the formula u = (B²) / (2μ₀). This concept is essential for calculating the energy density at the center of the loop, as it relates the magnetic field strength to the energy stored in that field.

Permeability of Free Space

The permeability of free space (μ₀) is a physical constant that describes how a magnetic field interacts with the vacuum of space. Its value is approximately 4π × 10⁻⁷ T·m/A. This constant is vital in the calculations involving magnetic fields and energy density, as it influences the strength of the magnetic field generated by the current in the wire.

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Related Practice

Textbook Question

(II) For the toroid of Fig. 30–26, determine the energy density in the magnetic field as a function of r(r₁ < r < r₂) and integrate this over the volume to obtain the total energy stored in the toroid, which carries a current I in each of its N loops.

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Textbook Question

(III) Determine the emf induced in the square loop in Fig. 29–52 if the loop stays at rest and the current in the straight wire is given by I(t) = (15.0 A) sin (2200 t) where t is in seconds. The distance a is 12.0 cm, and b is 15.0 cm.

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Textbook Question

(III) A toroid has a rectangular cross section as shown in Fig. 30–26. Show that the self-inductance is

$L = \frac{μ_{0} N^{2} h}{2 π} \ln \frac{r_{2}}{r_{1}}$

where N is the total number of turns and r₁, r₂ and h are the dimensions shown in Fig. 30–26. [Hint: Use Ampère’s law to get B as a function of r inside the toroid, and integrate.]

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Textbook Question

(II) (a) Determine the energy stored in the inductor L as a function of time for the LR circuit of Fig. 30–6a. (b) After how many time constants does the stored energy reach 99.9% of its maximum value?

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Textbook Question

The magnetic field inside an air-filled solenoid 38.0 cm long and 2.10 cm in diameter is 0.720 T. Approximately how much energy is stored in this field?

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Textbook Question

(II) Part of a single rectangular loop of wire with dimensions shown in Fig. 29–49 is situated inside a region of uniform magnetic field of 0.650 T. The total resistance of the loop is 0.250 Ω. Calculate the force required to pull the loop from the field (to the right) at a constant velocity of 3.40 m/s. Neglect gravity.

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