Can someone please explain the concept 'B' regarding magnetic fields? 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.
 A: "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!
A: 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.

