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

In summary, the equation of Lorentz force for the force acting on a moving charge in electric and magnetic field is:
  • #1
unplebeian
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1
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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  • #2
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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  • #3
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?
 
  • #4
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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  • #5
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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  • #6
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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  • #7
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.
 
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  • #8
## \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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  • #9
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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1. Why is the Lorentz force equation written as F = q(E + v×B) instead of F = qv×B?

The Lorentz force equation, F = q(E + v×B), takes into account both the electric and magnetic components of the force acting on a charged particle. The first term, qE, represents the electric force, while the second term, qv×B, represents the magnetic force. This equation is used because both electric and magnetic fields can act on a charged particle simultaneously, and the resulting force is a vector sum of these two components.

2. How does the Lorentz force equation account for the direction of the force?

The direction of the Lorentz force is determined by the cross product between the velocity of the charged particle and the magnetic field. This cross product results in a force that is perpendicular to both the velocity and the magnetic field. The direction of the electric force, on the other hand, is determined by the direction of the electric field, which is a vector quantity.

3. What is the significance of the Lorentz force equation in electromagnetism?

The Lorentz force equation is a fundamental equation in electromagnetism that describes the force experienced by a charged particle in the presence of electric and magnetic fields. It is used extensively in many fields, including particle physics, astrophysics, and engineering, to understand and predict the behavior of charged particles in various environments.

4. Can the Lorentz force equation be applied to all types of charged particles?

Yes, the Lorentz force equation can be applied to all types of charged particles, including electrons, protons, and other subatomic particles. It can also be used to describe the motion of larger charged objects, such as ions in a plasma or charged droplets in a cloud. However, the equation may need to be modified for particles with relativistic velocities or in extreme environments.

5. How does the Lorentz force equation relate to the concept of electromotive force?

The Lorentz force equation is closely related to the concept of electromotive force (EMF), which is the voltage generated by a changing magnetic field. The EMF is given by the cross product of the velocity and magnetic field, similar to the second term in the Lorentz force equation. This relationship is important in understanding electromagnetic induction and the generation of electricity in generators and motors.

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