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Ch 33: The Nature and Propagation of Light
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 33: The Nature and Propagation of LightProblem 22b
Chapter 33, Problem 22b

It is proposed to store 1.00 kWh = 3.60 × 106 J of electrical energy in a uniform magnetic field with magnitude 0.600 T. If instead this amount of energy is to be stored in a volume (in vacuum) equivalent to a cube 40.0 cm on a side, what magnetic field is required?

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First, understand that the energy stored in a magnetic field in a vacuum is given by the formula: \( U = \frac{1}{2\mu_0} B^2 V \), where \( U \) is the energy, \( B \) is the magnetic field, \( V \) is the volume, and \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \)).
Calculate the volume \( V \) of the cube. Since the side of the cube is 40.0 cm, convert this to meters (0.40 m) and use the formula for the volume of a cube: \( V = \text{side}^3 = (0.40 \text{ m})^3 \).
Rearrange the energy formula to solve for the magnetic field \( B \): \( B = \sqrt{\frac{2\mu_0 U}{V}} \).
Substitute the known values into the rearranged formula: \( U = 3.60 \times 10^6 \text{ J} \), \( \mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \), and the calculated volume \( V \).
Calculate the magnetic field \( B \) using the substituted values to find the required magnetic field strength to store the given energy in the specified volume.

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Key Concepts

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

Magnetic Energy Density

Magnetic energy density is the amount of energy stored per unit volume in a magnetic field. It is given by the formula u = B²/(2μ₀), where B is the magnetic field strength and μ₀ is the permeability of free space. Understanding this concept is crucial for calculating the energy stored in a given volume of space.
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Permeability of Free Space

The permeability of free space, denoted as μ₀, is a fundamental physical constant that describes how a magnetic field interacts with the vacuum. Its value is approximately 4π × 10⁻⁷ T·m/A. This constant is essential for calculating the energy density of a magnetic field in a vacuum.
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Volume of a Cube

The volume of a cube is calculated by cubing the length of one of its sides. For a cube with a side length of 40.0 cm, the volume is (0.40 m)³ = 0.064 m³. Understanding how to calculate the volume is necessary to determine the total energy stored in the magnetic field within this space.
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Related Practice
Textbook Question

Unpolarized light with intensity I0 is incident on two polarizing filters. The axis of the first filter makes an angle of 60.0° with the vertical, and the axis of the second filter is horizontal. What is the intensity of the light after it has passed through the second filter?

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Textbook Question

A beam of light strikes a sheet of glass at an angle of 57.0° with the normal in air. You observe that red light makes an angle of 38.1° with the normal in the glass, while violet light makes a 36.7° angle. What are the speeds of red and violet light in the glass?

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Textbook Question

A beam of unpolarized light of intensity I0 passes through a series of ideal polarizing filters with their polarizing axes turned to various angles as shown in Fig. E33.27. If we remove the middle filter, what will be the light intensity at point C?

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Textbook Question

The indexes of refraction for violet light (λ = 400 nm) and red light (λ= 700 nm) in diamond are 2.46 and 2.41, respectively. A ray of light traveling through air strikes the diamond surface at an angle of 53.5° to the normal. Calculate the angular separation between these two colors of light in the refracted ray.

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Textbook Question

The critical angle for total internal reflection at a liquid–air interface is 42.5°. If a ray of light traveling in the liquid has an angle of incidence at the interface of 35.0°, what angle does the refracted ray in the air make with the normal?

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Textbook Question

At the very end of Wagner's series of operas Ring of the Nibelung, Brünnhilde takes the golden ring from the finger of the dead Siegfried and throws it into the Rhine, where it sinks to the bottom of the river. Assuming that the ring is small enough compared to the depth of the river to be treated as a point and that the Rhine is 10.0 m deep where the ring goes in, what is the area of the largest circle at the surface of the water over which light from the ring could escape from the water?

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