Integral of magnetic field over the sphere

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SUMMARY

The integral of the magnetic field over a sphere can be expressed in two distinct scenarios: when all currents are inside the sphere, the relation is \(\int B \,dV = \frac{2}{3}\mu_0 M\), where \(M\) is the magnetic moment. Conversely, if all currents are outside the sphere, the relation becomes \(\int B \,dV = \frac{4}{3} \pi R^3 B(0)\), with \(B(0)\) representing the magnetic field at the center. These equations are derived from the principles outlined in "Classical Electrodynamics" by David J. Griffiths. The discussion raises the question of the magnetic field behavior when currents are located on the surface of the sphere, suggesting a need for further exploration of this unique configuration.

PREREQUISITES
  • Understanding of magnetic fields and magnetic moments
  • Familiarity with vector calculus and integrals
  • Knowledge of electromagnetic theory as presented in "Classical Electrodynamics" by David J. Griffiths
  • Basic concepts of current distribution in physics
NEXT STEPS
  • Explore the implications of surface current distributions on magnetic fields
  • Study the derivation of magnetic field equations in "Classical Electrodynamics" by David J. Griffiths
  • Investigate the application of the Biot-Savart Law in different current configurations
  • Learn about the mathematical techniques for evaluating integrals in electromagnetism
USEFUL FOR

Physicists, electrical engineers, and students studying electromagnetism who are interested in the behavior of magnetic fields in various current 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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