How can we transfer vector properties from one quantity to another?

Question

I am reading chapter "Moving Charges and Magnetism" in NCERT book. In the part talking about the force on a current-carrying wire in a uniform magnetic field, I read this strange transformation of vectors. I have highlighted the relevant parts.

Can someone explain how and why did that happen?

@ZeroTheHero isn't my question more about the transfer of vectors ? — Ankit, Oct 02 '21 at 17:46
There is no such thing as “transfer of vectors” and that phraseology is misleading. It so happens that $I d\vec \ell$ is defined to have the vector part over the line element rather than the current element, whereas for other geometries the vector part is on the current element, as per https://physics.stackexchange.com/a/325898/36194 — ZeroTheHero, Oct 02 '21 at 17:50

score 0 · Answer 1 · answered Oct 02 '21 at 18:31

$nq$ is the density of moving charges $\rho$. So, $$\mathbf j = \rho \frac{\mathbf{dx}}{dt}$$ If the density of current is constant over a straight wire: $$\mathbf j = \rho \frac{\mathbf{\Delta x}}{\Delta t}$$
Naming I as $$I = \frac{\Delta Q}{\Delta t} = \frac{\rho lA}{\Delta t}$$

and $\mathbf l = \mathbf{\Delta x}$, we get the final expression.

score 0 · Answer 2 · answered Oct 02 '21 at 19:29

0

The (j) is a unit vector indicating a direction (in an xyz coordinate system). The (l) is a length which has a magnitude. Assuming (l) is in the direction of (j), when you put them together you get a vector with a magnitude and direction.

answered Oct 02 '21 at 19:29

R.W. Bird

12,139

how is j a unit vector ? Isn't it the current density vector ? – Ankit Oct 03 '21 at 04:22
My mistake, but (j) is parallel to (l), so they do share the same direction. – R.W. Bird Oct 03 '21 at 15:47

How can we transfer vector properties from one quantity to another?

2 Answers2