# Why do magnetic field lines converge where the strength is strongest?

In most of the sites, they just say that "to determine the strength of the magnetic field, one must look at how many lines are present at a particular location." But my question is why? Why is this true? I am not able to get the intuition behind this.

And another question: The diagrams of magnetic fields are basically some curves, which are closer nearer the poles and away in the centre of the magnet. However, in physical space, is there any distance between these "lines"? I know that these lines are just a visualisation, but even then, magnetic field occupies the whole space(not physically, of course), but how just by looking at the diagram one can determine the strength of the field?

• The real reason is that the field obeys Gauss's Law, in free space. It is extremely shocking that this point is not stressed at all, in texts from Halliday Renick, to Griffiths, to Jackson, and many students end up thinking this is true of any vector field in general. The answer by NeuroFuzzy points to a reference, but again that doesn't seem like a very convincing argument. The only article in literature that has a proof of this that I've seen after looking for a brief while, is scitation.aip.org/content/aapt/journal/ajp/64/6/10.1119/1.18237 . – Aritro Pathak Nov 13 '16 at 20:11

The lines are indeed visualisations to represent a vector field.
At each point in space there is a magnetic field strength and a direction for that field.
The left hand diagram is such a representation for the magnetic field around a current carrying conductor with the current coming out of the screen.

If it was correctly drawn then the length of each of the arrows should be inversely proportional to the distance from the centre.
So this diagram gives you information about magnitude and direction.

The representation that you are perhaps more familiar with is thet in the right when the tangent to a field at a point gives the direction of the magnetic field line.
To illustrate the fact that the field is stronger near the conductor the concentric circles are drawn closer to one another.
So perhaps the second diagram does not have as much information on it as the first but it is significantly easier to draw.

However there diagrams are incomplete in that the magnetic fields are actually three dimensional and then the drawing of such diagrams becomes even more difficult.

Historically the magnetic flux density was the number of field lines per unit area and that is were the term flux (= flow) comes from with magnetic flux being the total number of lines.
You will still find lots of textbooks which are in esu, emu, cgs and Gaussian units from a time when there were also magnetic poles which followed an inverse square law just like Coulomb's law for electric charges.

So going back to your queries and the statement you made that they the magnetic field lines are visualisations and so you have some degree of artic licence with them provided you follow the simple properties:

• Start and finish on themselves although it is often much clearer if you have them starting on a North pole and finishing on a South pole.

• The arrow on a magnetic field line goes away from a North pole and goes towards a South pole or follows the right hand grip rule for currents.

• Magnetic field lines are in a state of tension. That is why a North pole attracts a South pole!

• Magnetic field lines never cross and repel each other. That is why two North poles repel one another!
• The closer the lines are to one another the stronger is the magnetic field (magnetic flux density).
• I'm not sure this gets to the heart of the question. In your example it's possible to draw uniform field lines, but in almost any other configuration (e.g. a simple magnet) it's not possible to draw uniform field lines; and the density of field lines indeed correspond to field strength... – lemon Apr 5 '16 at 16:55
• I understand what you mean particularly if you have magnetic field lines coming from, for example, magnetic poles. In some ways that is the beauty of magnetic field lines in that in a lot of examples you are forced to draw lines coming closer together or further apart and shows the insight that Faraday had when he wrote about and used his lines of force. I tried to give the idea that in lots of cases a visual picture of magnetic field lines is useful although there are no lines (or there are an infinite number of them). – Farcher Apr 5 '16 at 17:07

Perhaps the most convincing / intuitive answer comes from looking at some basic vector math.

Field is given as "flux", which we graphically equate with "lines per unit area".

Now if I actually wanted to know whether the field is stronger in one place than another, I would calculate the gradient, $\nabla B$. If the gradient points in a particular direction, I know that the field is stronger there.

So let's take a simple 2D diagram with diverging lines:

The gradient here is in the up-down direction. This tells us that the field is stronger in the lower part - where the field lines are closer together.

I don't know if that helps you tie the two concepts together...

The pdf J.D Callen, Fundamentals of Plasma Physics, chapter 3 defines $B=\sqrt{\vec{B}\cdot \vec{B}}$ and $\hat{b}=\vec{B}/B$, and proves that as you walk along a field line (arc length segment $d\ell$):

$$\frac{dB}{d\ell}=\hat{b}\cdot \nabla B=-B \nabla \cdot \hat{b}$$

(where the second equality holds from $\nabla\cdot(B \hat{b})=0$)

If the field lines are converging then $\nabla \cdot \hat{b}<0$ and so $B$ is increasing in magnitude, and if the field lines are diverging then $\nabla \cdot \hat{b}>0$ and so $B$ is decreasing in magnitude. So there's your vector calculus proof.

(This was an excerpt from my answer at https://physics.stackexchange.com/a/267588/12029)

• The fact that $\nabla \cdot \hat{B}<0$ implies the field lines are converging and vice versa, is true, but not very obvious. – Aritro Pathak Jan 20 '17 at 14:23

This property is a consequence of the divergence of the Magnetic field being being 0.

The important thing to note is you can start out by trying to draw as many fixed number of field lines on paper as you choose; the argument remains the same.

Consider any two “adjacent” field lines, both running from a region of high flux density to a region of low flux density. Consider a Gaussian box (admittedly I’m visualizing in 2 dimensions) whose sides are along the two adjacent field lines. One end of the pillbox is perpendicular to the field lines in the high flux density zone; the other end similarly perpendicular to the field lines, but this time, in the low flux density zone. The divergence of the magnetic field being 0, the flux through the narrow high flux area should equal the flux through the broad low flux area. This necessitates the relative lowering of field strength in the low flux region.

P.S: Because the curl of the field is 0 when no currents are flowing (as is the case here), the field strength along two adjacent lines would have to be almost similar, on the perpendicular plane. (Construct a suitable loop to see why this should be). This fact ensures that there is no arbitrary variation in the field strength on the perpendicular plane, and thus the argument of the previous paragraph works well.