Why isn't the magnetic field defined by the magnetic force on a particle moving through it?

A magnetic field describes the influence a charge (in motion) experiences. In other words, it is essentially a vector field that describes the force that a particle will feel at a given location. However, a moving charge experiences a magnetic force that is perpendicular the direction of its magnetic field and velocity. So my question is: why don’t we just define our magnetic field to be the vector field that is the direction (and magnitude) of the magnetic force that a particle experiences at a given location?

TL;DR: Why would we define the magnetic field to be perpendicular to the force a charge experiences, rather than just have it be the force that a charge experiences?

• Hi, you're more likely to get positive and constructive answers to your question if you select less...aggressive titles. The question itself isn't bad. – Richard Myers Apr 13 at 6:52
• I'm guessing you're a bit younger. In this site, the people expect each other to be 'formal' about the questions, because it will be that your question will be read by others in the future as well (assuming it is a good question, in this case I think it is one). So, it's best to just put the points directly (in a way which is acceptable to the community standards, you can search for posts discussing this on the meta of physics stack) – Buraian Apr 13 at 6:58

To make up a concrete situation consider a horseshoe magnet and a charge moving between the poles.

When you measure the force $$\vec{F}$$ acting on a charge $$q$$ moving through this magnetic field with various velocities $$\vec{v}$$ (e.g. in $$+x$$, $$-x$$, $$+y$$, $$-y$$, $$+z$$, $$-z$$ direction), then you get the following experimental results. Notice especially the $$+$$ and $$-$$ signs.

$$\begin{array}{|ccc|ccc|} \hline v_x & v_y & v_z & F_x & F_y & F_z \\ \hline +v & 0 & 0 & 0 & -qvB & 0 \\ -v & 0 & 0 & 0 & +qvB & 0 \\ 0 & +v & 0 & +qvB & 0 & 0 \\ 0 & -v & 0 & -qvB & 0 & 0 \\ 0 & 0 & +v & 0 & 0 & 0 \\ 0 & 0 & -v & 0 & 0 & 0 \\ \hline \end{array}$$

All these results above can be summarized like this: \begin{align} F_x &= +qv_yB \\ F_y &= -qv_xB \\ F_z &= 0 \end{align}

Note that, until now we didn't yet make any statement about how to define the direction of the magnetic field $$\vec{B}$$.

So my question is: why don’t we just define our magnetic field to be the vector field that is the direction (and magnitude) of the magnetic force that a particle experiences at a given location?

Looking at the results above, there is just no way to do this. You can't come up with a magnetic field vector $$\vec{B}$$ consistently having the same direction as the force $$\vec{F}$$.

The best thing you can achieve, is to rewrite the above with the help of the cross product. $$\begin{pmatrix}F_x \\ F_y \\ F_z \end{pmatrix} = q \begin{pmatrix}v_x \\ v_y \\ v_z \end{pmatrix} \times \begin{pmatrix}0 \\ 0 \\ B \end{pmatrix}$$

So you ended up with a vector $$\vec{B}$$ pointing in $$z$$-direction (i.e. perpendicular to the forces $$\vec{F}$$).

Because the force experienced by the particle isn't just dependent on its location, either in magnitude or direction.

The force depends on the particle location and it is perpendicular to its velocity. Thus if you change the velocity direction, the direction of the force changes. But if you want the magnetic field to be defined only as a function of position, then it must have a fixed direction at any point and cannot change just because the particle velocity has changed.

To put it another way, your magnetic force field could not be defined in the way you propose because it would be multiple-valued - at any position particles can have different velocities and therefore experience different forces.

The proper way to define a changing force vector that is always perpendicular to the velocity is then in terms of the vector product of velocity with another fixed vector - the magnetic field.

The magnetic field at a point is supposed to enable you to determine the force on a unit moving charge at THAT point.

You cannot define a magnetic field at a point "to be the vector field that is the direction (and magnitude) of the magnetic force that a particle experiences at a given location" because, the direction of the force is dependent on the charge and the vector of motion of the charge.

If you define the magnetic field according to the direction of magnetic force on a particular charged particle, that would not easily give you the answer to what would be the direction of force on another charged particle moving in a different vector.

By defining it the way it has been defined, it is easier to determine the direction of magnetic force on any charged particle moving in any vector

However, a moving charge experiences a magnetic force that is perpendicular the direction of its magnetic field and velocity. So my question is: why don’t we just define our magnetic field to be the vector field that is the direction (and magnitude) of the magnetic force that a particle experiences at a given location?

Very subtle point.

You see, the magnetic field 'decides' on how to push the particle based on how it enters and what charge it has. Let's fix the charge to be $$1C$$, so we can write the expression of force as:

$$\vec{F} = \vec{v} \times \vec{B}$$

Now, let's say that we 'rotate' our coordinate basis vectors such that the $$\vec{B}$$ is pointed along one our axes like say $$\hat{k}$$ theN:

$$\vec{F} = \vec{v} \times |B| \hat{k}$$

Or,

$$\vec{F} = |B| ( \vec{v} \times \hat{k})$$

And, we can write velocity of this particle as $$v= v_x \hat{i} + v_y \hat{j} + v_z \hat{k}$$, and hence the force is:

$$\vec{F} = (|B|) ( -v_x \hat{j} + v_y \hat{i})$$

Now, we see that the force at a point is determined by the velocity vector of the particle itself. So, suppose the velocity in $$j$$ direction was zero, then particle would only experience force in $$i$$ direction and similarly if velocity in $$i$$ direction was zero then $$j$$ direction would be the force.

Ultimately, the cross product is perfect because it easily always to say what we see in real life the easiest which is that the force experienced by a particle depends on what direction it was moving.

If I were to go down to the philosophical point of it, it is because the force is not determined solely by the external property but also how the internal property aligns with the external property