Question on the Lorentz force: Why is the force not F=q(v×B) = F=qv×qB

[unplebeian]

TL;DR Summary

Why is the charge not multiplied to the cross product

Background:

is the equation of Lorentz force for the force acting on a moving charge in electric and magnetic field.

For the magnetic field only it is : F=qv×B.

Question:
For magnetic field only why is the force not F=q(v×B) = F=qv×qB

 

[topsquark]

TL;DR Summary: Why is the charge not multiplied to the cross product

Background:

is the equation of Lorentz force for the force acting on a moving charge in electric and magnetic field.

For the magnetic field only it is : F=qv×B.

Question:
For magnetic field only why is the force not F=q(v×B) = F=qv×qB

You are only multiplying by q once, so
$q \textbf{v} \times \textbf{B}$

$= q ( \textbf{v} \times \textbf{B} )$

$= (q \textbf{v} ) \times \textbf{B}$

$= \textbf{v} \times (q \textbf{B})$

-Dan

 

[unplebeian]

Hi, Dan,
I'm sorry I didn't get it. That is a scalar multiplication so q should be multiplied to both. Generally a(bxc)= abxac.
Why are we multiplying only once?

 

[PeroK]

Generally a(bxc)= abxac.

This is wrong.
$$a(\mathbf b \times \mathbf c) = a\mathbf b \times \mathbf c = \mathbf b \times a\mathbf c$$You must be thinking of:
$$a(\mathbf b + \mathbf c) = a\mathbf b + a\mathbf c$$

 

[topsquark]

Hi, Dan,
I'm sorry I didn't get it. That is a scalar multiplication so q should be multiplied to both. Generally a(bxc)= abxac.
Why are we multiplying only once?

Is $2(3 \times 4 ) = (2 \cdot 3) \times (2 \cdot 4)$?

-Dan

 

[A.T.]

This is wrong.
$$a(\mathbf b \times \mathbf c) = a\mathbf b \times \mathbf c = \mathbf b \times a\mathbf c$$You must be thinking of:
$$a(\mathbf b + \mathbf c) = a\mathbf b + a\mathbf c$$

Easy to make mistake if in elementary school you learned the order of operations as "Dot ($\cdot$ and $\colon$) before stroke ($+$ and $-$)", because that's how the basic operators are written in your country.

 

[unplebeian]

Thank you, Dan. I thought about it graphically and it's evident that the scalar multiplication to both vectors prior to the cross product operation is incorrect. Rather take the cross product and then perform the scalar multiplication or simply any one vector like you suggested.

Thank you.

 

[Gavran]

$\vec F = q ( \vec E + \vec v \times \vec B )$

$q ( \vec E + \vec v \times \vec B ) = q \vec E + q ( \vec v \times \vec B )$ – the distributive property of scalar multiplication over the vector addition

$q ( \vec v \times \vec B ) = ( q \vec v ) \times \vec B = \vec v \times ( q \vec B )$ - the multiplication by a scalar property of the vector product (the multiplication by a scalar is not distributive over the vector product)

 

[PeroK]

$\vec F = q ( \vec E + \vec v \times \vec B )$

$q ( \vec E + \vec v \times \vec B ) = q \vec E + q ( \vec v \times \vec B )$ – the distributive property of scalar multiplication over the vector addition

$q ( \vec v \times \vec B ) = ( q \vec v ) \times \vec B = \vec v \times ( q \vec B )$ - the multiplication by a scalar property of the vector product (the multiplication by a scalar is not distributive over the vector product)