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

AI Thread Summary
The Lorentz force equation for a moving charge in a magnetic field is expressed as F = q(v × B). The discussion centers on why the charge (q) is not multiplied to both vectors in the cross product, as some participants mistakenly suggest. It is clarified that scalar multiplication applies only once to the entire vector product, not to each component separately. The distributive property of scalar multiplication over vector addition is correctly applied, emphasizing that the multiplication by a scalar is not distributive over the vector product. Understanding these properties is essential for accurately applying the Lorentz force equation.
unplebeian
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TL;DR Summary
Why is the charge not multiplied to the cross product
Background:
cb96d860cadff3d60e8ffb90b067b7f2b453c8e1
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
 
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unplebeian said:
TL;DR Summary: Why is the charge not multiplied to the cross product

Background:
cb96d860cadff3d60e8ffb90b067b7f2b453c8e1
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
 
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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?
 
unplebeian said:
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:
$$a(\mathbf b + \mathbf c) = a\mathbf b + a\mathbf c$$
 
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unplebeian said:
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
 
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PeroK said:
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.
 
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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.
 
## \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)
 
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Gavran said:
## \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)
:welcome:
 
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