Magnetism and Gauss’s law
What goes out must come in
Draw a closed surface around the north pole of a bar magnet, like a soap bubble that encloses just that end. Now count the field lines that cross the bubble.
Outside the magnet, lines leave the bubble, streaming away from the N pole. But follow them inside the magnet. There, the same lines enter the bubble, running from S to N. Every line that goes out also comes in.
To make this exact, divide the surface into tiny area elements. Each one is a vector ΔS, pointing outward. The magnetic flux through one element is ΔΦ_B = B · ΔS.
Where field lines leave the surface, B · ΔS is positive. Where they enter, it is negative. Because every magnetic field line is a closed loop, the positive and negative parts cancel exactly.
This is Gauss’s law for magnetism. The net magnetic flux through any closed surface is zero: Φ_B = Σ B · ΔS = 0, summed over all the elements of the surface.
It holds for a closed surface of any shape and size, anywhere: around a whole magnet, around one end of it, around a coil carrying current, or around empty space.
Compare Gauss’s law in electrostatics. There, the net electric flux through a closed surface is q/ε₀, where q is the charge inside. Draw a surface around the positive charge of an electric dipole: lines only come out. The flux is not zero.
Electric field lines begin and end on charges. Magnetic field lines never begin or end, because there is no magnetic charge for them to start on. Gauss’s law for magnetism is the statement that there are no magnetic monopoles.
So the simplest magnetic object is a dipole, or a current loop. Cut a magnet as often as you like: each piece is still a dipole, and the net flux through any closed surface is still zero.
Using Gauss’s law for magnetism
Gauss’s law for magnetism gives a quick test: can a drawing of field lines be a magnetic field at all?
Look at this picture: lines spreading out in every direction from one point, like the field of a point charge. Draw a small sphere around that point. Lines only leave it, so the net flux is positive. This can never be a magnetic field.
Here, lines stop suddenly in mid-air. Put a tiny closed surface around the place where they end. Flux goes in, and nothing comes out. Not allowed either.
Now the field around a long straight wire carrying current: circles around the wire. Any closed surface is crossed by each circle as many times going in as coming out. This one passes the test.
Worked example. A closed cube of side 10 cm sits in a uniform magnetic field of 0.5 T, pointing along x. Each face has area 0.1 × 0.1 = 0.01 m².
On the left face, B enters the cube, opposite to the outward ΔS. Its flux is −0.5 × 0.01 = −5 × 10⁻³ weber. On the right face, B leaves along ΔS: +5 × 10⁻³ weber.
The other four faces are parallel to B, so no lines cross them: zero flux. The total is −5 × 10⁻³ + 5 × 10⁻³ = 0, exactly as Gauss’s law says.
Second example. A closed surface encloses one end of a long solenoid. Inside the coil, 4 × 10⁻⁵ weber of flux enters the surface. Then exactly 4 × 10⁻⁵ weber must leave through the rest of it, where the lines spread out.
The unit of magnetic flux is the weber, Wb. One weber is one tesla times one square metre.
Remember the difference. For E, the net flux through a closed surface counts the charge inside. For B, there is nothing to count. The answer is always zero.
Try it: count the flux
- Drag the dashed closed surface. Amber dots mark lines going out, blue dots lines coming in.
- Put it around the N pole only (“Round N”). Lines leave on the outside but come back in through the magnet: net = 0.
- Try any size and position, even around the whole magnet. The net count is always zero.
- Switch to “Electric dipole” and enclose +q. Now lines only go out: the net is not zero, because a charge is inside.
- Cut the magnet in two and enclose one half. Still zero: there is no lone pole to enclose.
Around the north pole
A closed surface encloses only the N pole of a bar magnet. 3 × 10⁻⁴ Wb of magnetic flux leaves through the part of the surface outside the magnet. What is the net magnetic flux through the whole surface?
- 3 × 10⁻⁴ Wb
- 6 × 10⁻⁴ Wb
- Zero
- It depends on the strength of the pole inside
Answer
C. Field lines are closed loops. The 3 × 10⁻⁴ Wb that leaves outside the magnet enters the surface inside the magnet (where B runs from S to N). The net flux is zero, for any closed surface. There is no “pole strength” to enclose, as there is a charge in Gauss’s law for E.
Which one breaks the law?
Which statement is NOT consistent with Gauss’s law for magnetism?
- Magnetic field lines form closed loops.
- The net magnetic flux through a sphere drawn round a current loop is zero.
- Magnetic field lines start at the N pole and end at the S pole.
- Cutting a bar magnet in two gives two smaller magnets.
Answer
C. Outside a magnet the lines run from N to S, but they do not end at S: they continue inside the magnet from S to N and close on themselves. If lines ended on poles, a surface around one pole would have a net flux, and isolated poles would exist. Gauss’s law says Φ_B = 0 for every closed surface: no monopoles.
Summary
- Flux through a small area:
ΔΦ_B = B · ΔS - Gauss’s law for magnetism:
Φ_B = Σ B · ΔS = 0 (any closed surface) - Compare electrostatics:
Φ_E = q/ε₀ - Unit of flux:
1 Wb = 1 T m²
- The net magnetic flux through any closed surface is zero: every line that goes out also comes in.
- This is because magnetic field lines are closed loops, with no beginning and no end.
- Gauss’s law for magnetism means there are no magnetic monopoles. The simplest magnetic element is a dipole or a current loop.
- Use it as a test: a field pattern whose lines start or end somewhere cannot be a magnetic field.