It is proposed to store 1.00 kW•h = 3.60 * 10^6 J of electrical energy in a uniform magnetic field with magnitude 0.600 T. (b) 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?

Verified step by step guidance

1. First, understand that the energy stored in a magnetic field in a vacuum can be calculated using the formula for the energy density of a magnetic field, which is given by \( u = \frac{1}{2} \frac{B^2}{\mu_0} \), where \( B \) is the magnetic field strength and \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, T \cdot m/A \)).
2. Calculate the volume of the cube where the magnetic field will be stored. Since the side of the cube is 40.0 cm, convert this measurement to meters (0.40 m) and then calculate the volume using the formula for the volume of a cube, \( V = s^3 \), where \( s \) is the side length.
3. Using the total energy (3.60 * 10^6 J) and the volume calculated in the previous step, find the energy density \( u \) by dividing the total energy by the volume, \( u = \frac{E}{V} \).
4. Rearrange the formula for energy density to solve for the magnetic field strength \( B \). Solve \( B = \sqrt{2u \mu_0} \).
5. Substitute the values for \( u \) and \( \mu_0 \) into the equation to find the required magnetic field strength.

Key Concepts

Magnetic Energy Density

Magnetic energy density refers to the amount of energy stored in a magnetic field per unit volume. It is given by the formula u = (B^2)/(2μ₀), where B is the magnetic field strength and μ₀ is the permeability of free space. Understanding this concept is crucial for calculating how much energy can be stored in a given volume of space within a magnetic field.

Volume of a Cube

The volume of a cube is calculated using the formula V = a³, where 'a' is the length of one side of the cube. In this problem, the side length is given as 40.0 cm, which must be converted to meters for consistency in SI units. This volume will be used to determine how much energy can be stored in the specified magnetic field.

Energy Conservation in Magnetic Fields

Energy conservation in magnetic fields involves understanding how energy can be stored and transformed within a magnetic system. The total energy stored in the magnetic field must equal the energy input, which in this case is 1.00 kW•h. This principle allows us to relate the magnetic field strength required to store a specific amount of energy in a defined volume.