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 appropri- ate 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.
(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 appropri-
ate for the Earth's magnetic field.

Key Concepts

Boundary Conditions in Field Calculations

In scenarios where the field is only defined outside a certain region (such as outside a sphere), it is essential to correctly set the limits for integration to accurately determine quantities like total stored energy. This concept underlines the importance of applying appropriate spatial boundaries in physical integrations.

Integration of Inverse Power Laws

Many electromagnetic problems involve fields that decrease with distance according to a power law (such as the inverse square law or higher powers thereof). Integrating functions that involve inverse powers of the radial coordinate, especially when squared (as in computing energy density), requires careful handling of the resulting r-dependent terms.

Volume Integration in Spherical Coordinates

When calculating the total energy stored in a magnetic field, the integration is performed over the volume where the field exists. For problems with spherical symmetry, using spherical coordinates simplifies the computation due to the symmetry, with the volume element expressed as dV = r² sin? dr d? d?.

Magnetic Energy Density

This concept refers to the energy stored in a magnetic field per unit volume, typically given by the expression u = B²/(2??). This density allows one to calculate the total magnetic energy by integrating over the spatial region occupied by the field.

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