# Is there a better definition of magnetic field than this?

It may seem a trivial question but the definition of the magnetic field in everyday books is misleading. "It is the region or area around a magnetic material in which its magnetic force can be felt." It seems magnetic field is a physical area, thus its units must be meter square etc and not Tesla.

• Why do you think that is misleading? Also, the magnetic field is a volume... so m^3... – user207455 Jul 2 '19 at 8:37
• The word 'region' is misleading. It gives the impression that the field is necessary an 'area'. Why do we have its unit as Tesla if it is just an area or volume as you said? – Swami Jul 2 '19 at 8:38
• What you give there is not a "definition" of the magnetic field, but much rather a colloquial paraphrase. It's not clear what you're asking as even e.g. Wikipedia readily provides several actual definitions of the magnetic field. – ACuriousMind Jul 2 '19 at 17:20
• I agree that the question lacks a bit of research effort, but I find that this is a fairly common problem with (introductory) physics texts. There is not enough emphasis on proper definitions, so people start getting misleading ideas about various concepts which they only rely on intuition to understand. More advanced books are often rigorous, but by the time you get to this point, they will likely focus on the more advanced concepts. Therefore students can get confused. – Tob Ernack Jul 2 '19 at 17:45
• That book is doing a poor job of explaining physics. It might be best to find a better one. – G. Smith Jul 2 '19 at 22:27

A magnetic field is not a region or a volume, but a physical entity that can extend throughout a region or a volume. To be precise, it is part of an electromagnetic field, and the electromagnetic field is a physical entity whose primary observable property is that it exerts a force $${\bf f} = q ( {\bf E} + {\bf v} \times {\bf B})$$ on a particle of charge $$q$$ and velocity $$\bf v$$.

When I say "physical entity" here you should not think of it as a solid or fluid thing made of other stuff; I am just using the word "entity" in a general way to say "something which is there, and which has physical effects". We can define the electromagnetic field through its effects.

I said the magnetic field is part of an electromagnetic field because this is the correct way to see it in the more developed subject, but when you first meet it you will be introduced to cases where $${\bf E} = 0$$ and then of course you can talk about a magnetic field on its own.

I'm not sure where you got this definition of a magnetic field. A magnetic field classically only obeys two laws: The Maxwell equations from which you can determine possible configurations of the field given a current distribution AND the Lorentz force from which you can determine the magnetic field by looking at how the motion of moving charges are affected. You can in principle use either to "define" a magnetic field but I suspect the source you are reading is more to help lean you into the subject rather than provide a complete definition.

The magnetic field is an instance of what is known mathematically as a "vector field". This is a function from a vector space to another. For example, most physical vector fields would be functions from subsets of $$\mathbb{R}^3$$ to $$\mathbb{R}^3$$ (where the domain is a region of 3D euclidean space, and the codomain represents some vector quantity such as force, velocity, electric field, etc).

There is a region in which the magnetic field may have measurably nonzero magnitude (the set of all $$\mathbf{x}$$ such that $$|\mathbf{B}(\mathbf{x})| \neq \mathbf{0}$$), but this is not what we call the "magnetic field". It is more accurate to say that the magnetic field $$\mathbf{B}$$ is the function defined in that region. If you say "the magnetic field at a point $$\mathbf{x}$$", then you are referring to the value of this function at $$\mathbf{x}$$, that is, $$\mathbf{B}(\mathbf{x})$$.

So you can think of the magnetic field as a function that associates, to each point in space, a vector representing the magnetic field strength at that point. Intuitively, you can imagine a plot in 3d space where at each point, there is an arrow drawn.

Magnetic fields are actually functions of time in addition to space. So in general, the magnetic field is a function $$\mathbf{B}: \mathbb{R}^3 \times \mathbb{R} \to \mathbb{R}^3$$, where $$\mathbf{B}(\mathbf{x}, t)$$ is the value of the magnetic field at position $$\mathbf{x}$$ and time $$t$$. Maxwell's equations describe properties satisfied by $$\mathbf{B}$$ over space and time (as well as its relationship to the electric field $$\mathbf{E}$$). The Lorentz force law relates the magnetic field at a point to the force experienced by a moving charge at this point. You can define the magnetic field precisely as the vector field $$\mathbf{B}: \mathbb{R}^3 \times \mathbb{R} \to \mathbb{R}^3$$ satisfying Maxwell's equations and the (velocity-dependent part of) Lorentz force law.

In special relativity, the interpretation of the electromagnetic field is different. It is now considered a tensor field. At each point in spacetime there is a tensor representing the electromagnetic field. Tensors are generalizations of vectors. The electromagnetic field tensor combines the classical $$\mathbf{E}$$ and $$\mathbf{B}$$ fields. I am certainly not very knowledgeable in special relativity, but I am pointing this out to show that the meaning of electromagnetic field can change somewhat depending on the context.