Chapter 14: The Magnetic Field in Various Situations

The source material confirms that the divergence of the magnetic field (del dot B equals 0) is intrinsically satisfied when B is written as the curl of another vector. A significant point raised is that the vector potential is not uniquely defined; a different potential (A prime) that is equally satisfactory can be obtained by adding the gradient of any arbitrary scalar function (del psi) to A. For computational convenience in magnetostatics, the simplest choice is usually made by imposing the condition that the divergence of A must be zero (del dot A equals 0). The text proceeds to establish a direct mathematical method for finding the vector potential generated by known steady currents, deriving an integral equation for A which strikingly parallels the equation used to determine the scalar electrostatic potential (phi) from a charge distribution. This vector potential formulation is then applied to various physical scenarios, including calculating B for a long straight wire carrying current, and analyzing the vector potential outside a long solenoid carrying a uniform surface current. Furthermore, the chapter details the field created by a small current loop, revealing that its long-distance behavior is that of a magnetic dipole. The magnetic dipole moment (mu) is mathematically defined as the product of the current and the area of the loop, and the resulting magnetic field components are shown to have a form identical to those produced by an electric dipole. Finally, the process of calculating B from A using the curl operation leads directly to the formulation of the Biot-Savart Law, confirming the consistency between the vector potential approach and the established empirical law for magnetic fields produced by currents.