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I read the term field gradient in most of the article about magnetic field. I search it online but most of the explanation is about the math. I wonder in physics, what the gradient field really mean? To my understanding, I think a gradient field means the magnitude of field is position dependent, is that right?

I read an article online introducing the strength of the magnetic field of some objects. The Earth's magnetic field strength is about 0.5Gauss and a small magnetic bar (toy) produce about 5Gauss field. Let's take the magnetic toy as example. I wonder how to estimate the gradient of the field? From the math, it seems that we should estimate how does the field changed in a unit of distance, so for the bar producing field of 5Gauss, roughly how big is the gradient of that field?

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I think you are mixing two things: gradient and divergence.

The gradient is (normally) used when you have a scalar field, or function. A scalar field (or function) is when you associate a number to every point is space.

The divergence is (again, normally) used when you have a vector field, or function. A vector field (or function) is when you associate a vector (let's say x,y and z) to every point in space.

The magnetic field is a vector field, so you would normally just look at the divergence of that field. That being said, the vectorial gradient is something that exists, but is at a much higher level (https://math.stackexchange.com/questions/156880/gradient-of-a-vector-field).

Now, if you meant "what is the divergence of the magnetic field?" The answer is 0, because it's always 0. This is actually an experimental truth and if you want to understand this, i'll refer you to this nice explanation: Why is the divergence of a magnetic field equal to zero?

EDIT: Thanks to @NeuroFuzzy for the pointer. So, to complement my answer, if by gradient you mean variation over a small distance of the magnetic field, then i'll compare three situations: 1. For an infinite wire (think of your power lines), the magnetic field will run around the wire following the right hand rule. The intensity of the field will decrease following ~ 1/distance. In this case, you would say that the gradient is not very high because it doesn't vary a lot with distance.

  1. For a loop of current, the field at a distance from the center of the loop is a complicated equation. To get an intuitively pleasing result, you need to approximate how the field behaves at a good distance. This gives a field that derceases following ~ 1/distance^3, which means that it has a very strong gradient - it varies a lot.

  2. Lastly, my trick for visualizing a magnet is to think of it as a multitude of infinitely small current loops. Admitting this, and doing some (rather tough) math, you'll eventually find a formula (see eq. 8) which, for the same approximation as in case 2, give a variation of ~ 1/distance^3 again.

So, the final answer, would be to say that the gradient of your magnet is stronger than that of a power line - it varies more!

Hope it helps!

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  • $\begingroup$ Hi @victorbg, "gradient" can mean many different things. If someone is reading about a "gradient in the magnetic field", they probably mean "gradient" in the more general/english sense of the word, not the "turns a scalar field to a vector field" sense of the word. So, for example, if someone had an experiment with a "large magnetic field gradient", I would think something like the magnets involved in the Stern-Gerlach experiment en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment . Also relevant: en.wikipedia.org/wiki/… $\endgroup$
    – user12029
    May 19 '15 at 22:29
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Where were you reading this stuff?

It is certainly true the if the field has a gradient then the magnitude will be position dependent. Sometimes, however, a popular publication with confuse "field" with "potential" - so you need to watch for that.

To find the gradient of the 5Gauss toy - you should first look up the definition of a "Gauss". You'll find those field strengths are determined as an average at a particular distance. How the field changes in space depends on the details of how the field is generated. Magnetic fields can be quite complicated because of the poles.

i.e. strictly speaking there is not enough information from the magnitude of the field by itself.

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You calculate the gradient of a scalar field and get a vector field as a result.

  • scalar field means for every point in space, the field has a value.
  • vector field means for every point in space, the field has a vector, i.e. a value and a direction.

To get an intuitive understanding, imagine the surface of the earth as a scalar field. On a map for example, you see that for every point on it, there is a height value of the terrain.

What happens if you place a ball on a position? If it is placed in a valley, it won't move. If it is placed on a slope of a hill, it will roll down the hill. Depending on where you place it, the ball will have a certain velocity (a speed value and a direction in which it will move). If you calculate the velocity for every point on the map, you create a vector field from the scalar field that is the map. You calculated the gradient.

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