Magnetism and Gauss's Law

Statement of the law

Gauss's law for magnetism: the net magnetic flux through any closed surface is always zero.

$$\phi_B = \sum_{\text{all area elements}} \vec{B}\cdot\Delta\vec{S} = 0$$

• $\Delta\vec{S}$ is a small outward-pointing vector area element of the closed surface $S$, and $\vec{B}$ is the field at that element. The flux through one element is $\Delta\phi_B = \vec{B}\cdot\Delta\vec{S}$; summing over all elements of any closed surface gives zero.
• In integral form, $\oint_S \vec{B}\cdot d\vec{S} = 0$.
• Flux is a scalar, measured in weber, with $1\ \text{Wb} = 1\ \text{T m}^{2}$.

Why the flux is zero — the argument to write out

• Magnetic field lines are continuous and form closed loops; they have no beginning and no end.
• So for any closed surface, every line that enters it must also leave it. The number of lines leaving is exactly balanced by the number entering.
• Hence the net outward flux is zero. Drawing the Gaussian surfaces labelled $i$ and $ii$ around parts of a bar magnet, or around parts of a current-carrying solenoid, both show this: whichever way you cut it, lines in equal lines out.

Diagram to draw: a bar magnet with its closed field lines, and two dashed closed curves — one enclosing the N pole only, one enclosing the whole magnet. Mark, for each, that every line crossing outwards is matched by a line crossing inwards through the magnet's body, so $\phi_B = 0$.

Significance (the part that carries the marks)

• The law establishes that isolated magnetic poles — magnetic monopoles — do not exist. This is why a bar magnet cannot be broken into a separate N pole and S pole.
• There are no sources or sinks of $\vec{B}$. Electric field lines begin on positive charge and end on negative charge; magnetic field lines do neither.
• The simplest magnetic element is a magnetic dipole, or equivalently a current loop. All magnetic phenomena can be explained by an arrangement of dipoles and current loops.
• It explains why field lines are continuous closed curves — that continuity is a consequence of the law, not an extra assumption.

Contrast with Gauss's law of electrostatics

• If magnetic monopoles did exist, the law would be modified to match the electrostatic one:

$$\oint_S \vec{B}\cdot d\vec{S} = \mu_0\, q_m$$

where $q_m$ is the magnetic charge enclosed by $S$. Only quote this when a question explicitly asks how the law would change.

Using the law to spot a wrong field-line diagram

• Lines radiating outwards from a point: impossible for $\vec{B}$, because the net flux through a small surface round that point would be non-zero. That picture is the electric field of a long positively charged wire; the correct magnetic picture round a straight conductor is concentric circles.
• Two lines crossing: impossible for both $\vec{B}$ and $\vec{E}$ — the field direction at the crossing would be ambiguous.
• A closed loop of static magnetic field line enclosing empty space: wrong. A closed magnetostatic field line must enclose a region across which a current passes. Electrostatic field lines can never form closed loops at all.
• Lines forming closed loops entirely inside a toroid: correct, since each loop encloses current.
• Solenoid lines drawn perfectly straight and fully confined at the ends: wrong; they must curve out at the ends and close, otherwise Ampere's law is violated.
• Lines inside and outside a bar magnet forming closed loops: correct. Note the direction inside — S to N. Not all lines emanate from the N pole; around either pole the net flux is zero.
• Lines all emerging from one shaded plate and terminating on another: an electrostatic pattern between charged plates, impossible for $\vec{B}$ because the flux out of a surface round the upper plate would not be zero.
• Field between two pole pieces drawn perfectly straight to the edges: wrong; fringing at the ends is inevitable, for magnetic and for electric fields alike.

Related points the board has asked

• Magnetic field lines show the direction along which a small magnetised needle aligns; they do not represent lines of force on a moving charge, because the magnetic force $q\vec{v}\times\vec{B}$ is always normal to $\vec{B}$.
• A bar magnet exerts no torque on itself due to its own field, and one element of a current-carrying wire feels no force from its own field — although it can feel a force from other elements of the same wire (zero in the special case of a straight wire).
• A system with zero net charge can still have a magnetic moment, since the moments of the atomic current loops need not cancel. Paramagnetic atoms are exactly this case.