What is magnetic flux density I can't really understand what magnetic flux density really is, why when there are two opposite magnetic flux lines coming into an area they cancel each other? They are not a force to do so even if they are vectors I can't understand it. What are magnetic flux lines what is their importance. I'm really confused.
 A: First we should understand what magnetic flux $\Phi_B$ is. It has the units Wb (equivalent to $V s^{-2}$) and can be defined by the following surface integral:
$$\Phi_B = \int \int_S \textbf{B} \cdot d \textbf{S}$$
Where B is often called the magnetic field and has units T (equivalent to $kg \space s^{-2} A^{-1}$). It is more strictly known as magnetic flux density.
Magnetic fields are best thought of as having field lines, which are a handy way of visualising any kind of vector field. In short, field lines can visually represent the field's influence and in what direction the force acts on interacting objects at a particular location. Magnetic flux $\Phi_B$ can be thought of as a measure of how many magnetic field lines pass through a given surface. The surface can be either be real or imagined.
If magnetic flux $\Phi_B$ is the number of magnetic field lines that pass through a given surface, magnetic flux density B is the normalised number of magnetic field lines passing through, i.e. the number of magnetic field lines passing through per unit surface.
As you may be able to see from the equation given above, magnetic flux density can be thought of as magnetic flux divided by the area of the surface.
The relationship between magnetic flux and magnetic flux density is similar to the relationship between mass of an object and that object's density (although this example considers 3 dimensional volumes, whereas magnetic flux and magnetic flux density are only concerned with 2 dimensional surfaces).
Why it is useful: Knowing the magnetic flux for a given surface, such as a sheet of metal being magnetised, is useful, however magnetic flux will change depending on what size the sheet is. The magnetic flux density, however, applies no matter what size the sheet of metal is, as long as distance from the magnetic field source is kept the same. So using magnetic flux density, you can calculate magnetic flux arbitrarily.
A: Magnetic flux lines are a visualization tool. IMHO, don't worry about them too much. They are as 'real' as the convention for colouring positive charges red, and negative charges blue, when creating illustrations.
Magnetic field on the other hand, i.e. a function that gives you a vector for any position (and time) is more fundamental.
The reason magnetic fields add, or subtract, depending on the sign, is the linearity of Maxwell's equations (https://en.wikipedia.org/wiki/Maxwell%27s_equations) in vacuum. In non-linear environments, e.g. ferro-magnetics, this is no longer true.
Final question, why are you more comfortable with force than magnetic field? Both are mathematical constructions. Moreover, apart from gravity, all forces you feel are mediated by electromagnetic fields.
A: This question is asked all the time by beginners and there's not really a satisfying answer. There is no good mechanical analog with which to use to get around the fact that the human brain was not meant to intuit something like this.
It also more obviously brings to light the stark difference between physical reality and a model that just is able to predict physical reality.
You should deeply examine yourself and think about why lines of force make sense to you, but not some other lines. In the end, you should discover that the only reason you do not ask the same thing about lines of force is because your direct everyday experience with force has caused you to become so used to it that you stop questioning or thinking about what it is, whereas magnetic and electric flux density is new and unfamiliar so you do not take anything for granted.
Personally, I visualize the magnetic field intensity, H, as how bright the lines are, and visualize the magnetic field density as how close together the lines are.
