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We have the formula for the Lorentz force $$\textbf{F} = q \space(\textbf{E} + \textbf{v} \times \textbf {B})$$

This is a simple formula you learn in high school, but I'm interested to self-study electromagnetism and I found out a different notation for this formula:

$$ \textbf{F}(\textbf{r}, \dot{\textbf{r}} , t,q) = q[\textbf{E}(\textbf{r}, t) + \dot{\textbf{r}} \times \textbf{B}( \textbf{r}, t)]$$

I want to know where I can study this type of notation, as I've never encountered it during high school. I'm guessing the LHS needs to include every unit on the RHS, but I don't understand for example why we have $\textbf{E}(r,t)$, what is the significance of the position vector and time, and how can you know?

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I guess the second equation you wrote deals with variable electric and magnetic field. It shows how the force on a particle of charge q,at a time t,velocity $r^.$ and position r is dependent on the electric and magnetic field at that point and time – Rajath Krishna R Nov 21 '13 at 15:37
Leonard Susskind in his lectures of Theoretical Minimum deals with these types of notations. – Rajath Krishna R Nov 21 '13 at 15:41
up vote 3 down vote accepted

The explanation does not lie in the vectorial nature of the quantities at hand, but rather in the fact that they can be viewed as functions of several variables. Just as a scalar function $f$ can depend on a variable $x$ and be denoted $f(x)$, or it can depend on two variables $x$ and $y$ and be denoted $f(x,y)$, and similar for more variables, a vector can be a function in just the same way and depend on as many quantities as you like. What they are in your concrete example depends on the physical system you want to describe.

The Lorentz force $\mathbf{F}$ depends on the position $\mathbf{r}$ (which is not the unit vector), the velocity $\mathbf{v}=\mathbf{\dot{r}}$, time $t$ and charge $q$. Thus, in order to make this explicit, we denote it $\mathbf{F}(\mathbf{r},\mathbf{\dot{r}},t,q)$. Since we have an equation for this force, we expect all these arguments to occur on the right hand side as well. Velocity and charge appear as multiplicative quantities, position and time as arguments of two other (vectorial) functions: the electric field $\mathbf{E}(\mathbf{r},t)$ and the magnetic field $\mathbf{B}(\mathbf{r},t)$.

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Thanks, and you are correct, I accidentally confused the position vector with the unit vector. – user30117 Nov 21 '13 at 15:53

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