Assume the magnitude of the magnetic field outside a sphere...

Assume the magnitude of the magnetic field outside a sphere of radius $R$ is $B=B_{0}(R / r)^{2},$ where $B_{0}$ is a constant. (a) Determine the total energy stored in the magnetic field outside the sphere. (b) Evaluate your result from part (a) for $B_{0}=5.00 \times 10^{-5} \mathrm{T}$ and $R=6.00 \times 10^{6} \mathrm{m},$ values appropriate for the Earth's magnetic field.

Key Concepts

Magnetic Field Energy Density

This concept explains that any magnetic field carries an associated energy that is quantified by its energy density. The energy density in a region with a magnetic field is given by the expression u = B²/(2??), where B represents the magnetic field strength and ?? is the permeability of free space. This formula is fundamental in calculating the total energy stored in the magnetic field over a specified volume by integrating this density across that volume.

Volume Integration in Spherical Coordinates

When a physical problem exhibits spherical symmetry, it is often most convenient to perform integrations using spherical coordinates. In this coordinate system, the volume element is expressed as dV = r² sin(?) dr d? d?. Using these coordinates simplifies the integration process, especially when the field or other physical quantities depend solely on the radial distance, allowing for separable integrations over the angles and the radius.

Inverse Square Dependence of the Magnetic Field

The inverse square behavior, expressed as B ? (R/r)², indicates that the magnitude of the magnetic field decreases with the square of the distance from the source. This kind of dependence often arises in systems exhibiting spherical symmetry, and it greatly simplifies the subsequent integration process when determining quantities like total energy. Recognizing this dependence helps in simplifying the mathematical treatment of the problem by reducing the complexity of the integrals involved.

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