Integral of magnetic field over the sphere

AI Thread Summary
When all currents are inside a sphere of radius R, the integral of the magnetic field over the volume equals 2/3μ₀M, where M is the magnetic moment. Conversely, if all currents are outside the sphere, the integral equals 4/3πR³B(0), with B(0) being the magnetic field at the center. The discussion raises a question about the scenario where currents are located on the surface of the sphere and whether both previous relations can hold simultaneously. Participants are encouraged to explore this situation further to derive insights. The conversation emphasizes the need for analytical exploration of magnetic field behavior in different configurations.
hokhani
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If all the currents were inside a sphere with the radius R, then we would have \int B \,dV= 2/3\mu_0 M where M is magnetic moment of all the currents and B is magnetic field. If all the current were outside the sphere, then we would have\int B \,dV= 4/3 \pi R^3 B(0) where B(0)is magnetic field at center of the sphere (Both the relations above are derived in Jackson).
Now, how about the situation in which all the currents were on the surface of the sphere? Can one say that both the two relations above are hold simultaneously?
 
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Why not work it out and see?
 
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