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

Name: Problem 29, Chapter 32: Physics for Scientists and Engineers with Modern Physics
Uploaded: 2020-08-03T08:53:40-08:00
Duration: 5 min 24 s
Channel: Keshav Singh
Description: Assume the magnitude of the magnetic field outside a sphere of radius R is B=B0(R / r)^2 , where B0 is a constant. Determine the total energy stored in the magnetic field outside the sphere and evaluate your result for B0= 5.00 ×10^-5 T and R=6.00 ×10^6 m, values appropriate for the Earth's magnetic field.

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. Determine the total energy stored in the magnetic field outside the sphere and evaluate your result 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.

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Key Concepts

Magnetic Energy Density

This concept refers to the amount of energy stored per unit volume in a magnetic field, typically given by B^2/(2??) in free space. It is fundamental for calculating the total energy contained within a magnetic field by integrating over the region of interest.

Spherical Volume Integration

When the magnetic field exhibits spherical symmetry, the total energy is calculated by integrating the energy density over the volume using the spherical coordinate volume element, 4?r²dr. This method simplifies the integration process by exploiting the symmetry of the scenario.

Radial Variation of the Magnetic Field

Understanding how the magnetic field varies spatially, especially when it decays as a function of distance (such as an inverse-square law), is crucial. This knowledge allows one to substitute the radial dependence into the energy density expression, thereby determining how the energy contribution diminishes with increasing distance from the source.

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