Physics · Ch 12 — Magnetism
Gauss' Law of Magnetism
Gauss' Law of Magnetism
Gauss' law of electrostatics states that the net electric flux through any closed (Gaussian) surface is proportional to the net electric charge enclosed by that surface. Gauss' law for magnetic fields makes an analogous-looking but crucially different statement: the net magnetic flux through ANY closed Gaussian surface is always exactly zero, .
This can be seen by comparing three cases side by side: (a) a bar magnet, (b) a current-carrying finite solenoid (whose external field pattern looks just like a bar magnet's), and (c) an electric dipole, each with the cross-sections of two closed three-dimensional Gaussian surfaces, labelled (i) and (ii), superposed on their field-line diagrams. For surface (i), which does not enclose either pole (or either charge), the number of lines entering trivially equals the number leaving in all three cases -- unsurprising for any closed surface with nothing of interest inside it. For surface (ii) in case (a) or (b), which is drawn so as to enclose just the north pole (or one end) of the magnet or solenoid, one might at first expect a nonzero outward flux -- but because even the thinnest possible slice of a real bar magnet still contains BOTH a north-type region and a south-type region (poles are never isolated), the enclosed net "magnetic charge" remains zero, and the flux again works out to zero, with equal numbers of lines still entering and leaving.
Case (c), the electric dipole, is entirely different: because its field lines begin on the positive charge and end on the negative charge WITHOUT closing into loops, a surface (ii) enclosing only the positive charge genuinely does have more lines leaving than entering -- a real nonzero net outward flux, exactly as required by Gauss' law of electrostatics, since that surface really does enclose a net positive charge. …
What this figure shows. A bar magnet is drawn with its external field lines curving from the N pole around to the S pole, forming closed loops that continue through the body of the magnet. Two cross-sections of closed three-dimensional Gaussian surfaces are superposed: curve (i), a small loop that does not enclose either pole (showing equal numbers of field lines entering and leaving), and curve (ii), a loop that encloses just the N pole region (yet still shows equal lines in and out overall, since even that thin enclosed slice of the magnet carries b …
What this figure shows. A finite solenoid carrying current I is drawn with external field lines that curve from one end (behaving like a N pole) around to the other end (behaving like a S pole), closing through the inside of the solenoid, exactly matching the bar magnet's field pattern in Fig. 12.4(a). The same two Gaussian cross-section curves (i), enclosing no pole-like end, and (ii), enclosing one pole-like end, are superposed, again showing zero net flux through either closed s …
What this figure shows. An electric dipole, with a positive charge and a negative charge, is drawn with its field lines running OPEN -- starting on the positive charge and terminating on the negative charge, never closing into loops (unlike the magnetic cases in Figs. 12.4(a) and (b)). The same two Gaussian cross-section curves (i) and (ii) are superposed; curve (ii), which encloses the positive charge alone, now shows a genuine nonzero NET OUTWARD flux (more lines leaving than entering), consistent with -- the key contrast that motivates Gauss' law of magnetism …