# A charged particle, a magnetic field and the particle's path

Quoted below is a question & the provided answer:

## Question:

Describe the path of the $$\alpha$$ particles in the space between the magnetic poles.

• curved path stated or drawn
• path at right angles to magnetic field
• into paper

I've understood the first and the last points, but the second one is quite new to me, and unexplained by my textbook (written for high school sophomores studying introductory-level atomic physics). A search on Google led me to the following paragraph, which reiterates the second statement provided by the marking scheme:

A charged particle experiences a force when moving through a magnetic field. Since the magnetic force is perpendicular to the direction of travel [...]

However, the explanation as to why this occurs was not very well conveyed & seemed highly esoteric for my understanding. Could someone please shed some light on this topic, or direct me to places where I could learn more about this?

Why is the direction of the magnetic force on a moving charge perpendicular & why has it been said "path at right angles to magnetic field" in the marking scheme? [Please do describe the mathematics behind it, if it aids the explanation]

• By the last comment in bold, do you mean...'perpendicular to the magnetic field', this is true if there is no original component of motion parallel to the field...or did you mean 'Why is the force on a charged particle...'? Oct 3 '21 at 16:19
• That's why we need mathematics to understand what is going on $$\mathbf{f} = q\left(\mathbf{E}+\mathbf{u}\boldsymbol{\times}\mathbf{B}\right)$$ Oct 3 '21 at 21:29
• I wouldn't mind the mathematics, as long as it is explained! @Frobenius Oct 4 '21 at 2:25
• Also, for those who had voted to close my "homework-like" question: no, I'm neither looking for answers to the question (which I already have) nor am I asking to check my work (which I have not even presented). What I'm looking for here is merely just an explanation as to why the force is perpendicular to the magnetic field. True, I did quote a question, but that's just to supplement the question! Oct 4 '21 at 2:27

Flemings Left Hand Rule can be used to find the direction of the force on the positively charged $$\alpha$$ particle

and it gives a force into the paper, it's always perpendicular to both the magnetic field and the direction of travel. So the particle will start curving in circular motion away from us into the paper.

The direction of travel of the particle during this circular or part circular motion is always at right angles to the magnetic field.

The mark scheme was looking for a description of in which plane the curved path lies. The circle or part circle is between the poles as if we put a thin circular disk between and the particle travelled along the rim. The direction of such a motion is always perpendicular to the magnetic field, even when it gains a component of motion into the paper.

Mark schemes are sometimes unfair, as the candidate has to kind of guess what kind of point or phrase is needed, but unfortunately that's how they mark certain papers at the moment.

There is not really a simple way to explain "why" the force behaves as it does, which is to say why the force will be perpendicular to both the particle velocity and the magnetic field vectors. This is an experimentally observed fact.

If you want a peek into a slightly deep explanation, however, I can offer this one. Perhaps it is less "strange" to you that electric charges experience an electric force of the form $$\mathbf{F}_e = q \mathbf{E}$$. Once you study relatively, you will learn that electric and magnetic fields are intimately related, and given the electric force quoted above, it is necessary that a magnetic force behave like $$\mathbf{F}_m = q \mathbf{v} \times \mathbf{B}$$ in order to maintain consistency between different reference frames. The cross product in the magnetic force expression is what guarantees the force is perpendicular to both $$\mathbf{v}$$ and $$\mathbf{B}$$.

Basically, the answer is that: both the electric and magnetic forces originate from a single electromagnetic force, but it requires a deeper study of relativity to understand fully the connection. Hopefully this example at least shows you that there is some deeper understanding to come if you continue to study physics!