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I can't seem to understand the concept 'B'. I have done a great deal of research regarding magnetism and magnetic fields but I seem to have just complicated the matter further.

What I understand so far is that magnetic fields are produced by moving charges & that the magnetic field is a vector field. I understand that a compass can give us the direction of a magnetic field at any point.

Now my problem arises. I know that B is called the magnetic field and, from the Lorentz force law, B is the force applied to a moving unit charge in the unit Tesla. However, isn't B also regarded as the magnetic flux density (the amount of flux per unit area)? If this is so, then how can B represent both the magnetic field and the magnetic flux density? Is the magnitude of the magnetic field and magnetic flux density the same thing?

Sorry for the length of this, thanks in advance.

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2 Answers 2

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"If this is so, then how can B represent both the magnetic field and the magnetic flux density? Is the magnitude of the magnetic field and magnetic flux density the same thing?"

Yes!

These days the Lorentz force is usually used to define what we mean by the $\vec{B}$ vector: if a charged particle with velocity $\vec{v}$ experiences a force given by$$\vec{F}=q\vec{v}\times \vec{B},$$then $\vec{B}$ is the local magnetic field strength (officially, the magnetic flux density).

We define the flux passing through an area, S, as the surface integral of $\vec{B}$ across that area. Thus $$\Phi=\int_{S}\vec{B}\cdot \mathrm d\vec{S}.$$

$\Phi$ is especially important for electrical engineers designing motors, generators and transformers, and they may well regard it as the quantity of primary interest, and $\vec{B}$ as a derivative quantity – in which case it's natural to think of the last equation as defining $\vec{B}$ as a flux density! Even though physicists don't usually define $\vec{B}$ in this way, it's clearly not wrong to regard $\vec{B}$ as a flux density, and this point of view gives $\vec{B}$ its name!

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    $\begingroup$ Isn't it necessary for $\vec{dS}$ to be parallel to $\vec{B}$ for the flux to be the integral of all of the B field? $\endgroup$
    – JMLCarter
    Commented Jun 25, 2018 at 22:12
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    $\begingroup$ @JMLCarter: There is a dot product between the vector $\vec{B}$ and the vector area $d\vec{S}$ in the integral, which picks up exactly the normal component. $\endgroup$
    – NickD
    Commented Jun 25, 2018 at 22:14
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    $\begingroup$ Yes, but the direction of $\vec{B}$ is at right angles to the force experienced by the moving charge, and at right angles to the direction of motion of the charge. The magnitude of $\vec{B}$ is $B=\frac{F}{q\ v\ }$ in which $v$ is the charge's speed and $q$ is its charge, if the particle's direction of motion is such as to maximise the force (that is it is moving at right angles to the field). $\endgroup$
    – Philip Wood
    Commented Jun 26, 2018 at 17:05
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    $\begingroup$ I wouldn't say "by definition". As I tried to explain, I think most physicists would choose to define $\vec{B}$ in terms of the Lorentz force. Flux, $\Phi$, is then defined as $\Phi=\int_{S} \vec{B}.d \vec{S}$. So this last equation is defining $\Phi$ in terms of $\vec{B}$, not the other way round. But it may not always have been like this. I believe that at one time, $\Phi=\int_{S} \vec{B}.d \vec{S}$ was used as the definition of $\vec{B}$, and that even now, engineers may do so. Hence the name "flux density". Hope this helps. $\endgroup$
    – Philip Wood
    Commented Jun 29, 2018 at 8:30
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    $\begingroup$ I've just realised that what may be worrying you is the discrepancy between "at a point" and "per unit area". What we mean by "per unit area" is the limit of $\frac{d\Phi}{dA}$ AS $A$ APPROACHES ZERO, when the area is orientated for maximum flux to pass through it, that is orientated at right angles to the direction of $\vec B$. Therefore the "flux per unit area" definition is also for a point. $\endgroup$
    – Philip Wood
    Commented Aug 2, 2019 at 15:56
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There is a connection between two quantities relating to magnetism

$$\textbf B= \mu \textbf H$$.

$ \textbf H$ is the magnetic field strength and has units of $\rm A\, m^{-1}$ $(\rm ampere\, metre^{-1})$

$ \textbf B$ is the magnetic flux density and has units of $\rm T$ $(\rm tesla)$

$\mu$ is the permeability of the material and has the units $\rm H\,m^{-1}$ $(\rm henry\, metre^{-1})$

The problem you are having is possibly due to two factors.

  • Often in common usage rather than say the longer “magnetic flux density” the term “magnetic field” is used.
  • The theory of magnetism developed via the concept of magnetic field lines and magnetic poles whose strength was related to the number of magnetic field lines. The term “magnetic flux density” referred to the number of these magnetic field lines which passed through unit area with the word “flux” indicating a “flow” of these magnetic field lines through the area. This is what your $\textbf B$ stands for as is used in the Lorentz force formula.
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  • $\begingroup$ Could you please explain where the quantity H comes in and why we use it when we already use the quantity B for magnetic fields? Is H analogous to E for electric fields (where E is the force per unit charge); or is B analogous to E (as it's the force per unit moving charge). $\endgroup$
    – S H
    Commented Jun 26, 2018 at 7:18
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    $\begingroup$ Have a look at this webpage hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfield.html $\endgroup$
    – Farcher
    Commented Jun 26, 2018 at 7:20

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