An apparent contradiction between Fleming's rules

[brotherbobby]

TL;DR

Fleming's left hand rule gives the force experienced by a current carrying conductor to point along the thumb, where the middle finger points along the current and the index along the (magnetic) field. The Lorentz force on a charge moving in a magnetic field is given by Fleming's right hand rule. Taking a moving charge in a straight line to be (crudely) as current, don't the two rules give opposite directions of force on moving charges?

Situation : I have drawn the image to the right. A current $I$ flows "down" along a wire in a magnetic field $\vec B$ directed to the "right". By Flemings's Left Hand Rule (LHR), the wire should experience force out of the page, as shown by the green bullet $\color{green}\bullet$. However, the current can be crudely approximated by a charge $q$ moving "down" with some velocity $\vec v$. The force on the charge is given by $\vec F=q(\vec v\times\vec B)$.

By using Fleming's Right Hand Rule (RHR) for the same situation, remembering that this is a vector "cross" product, the force on the charge comes out to be into the page, as shown by the mark $\color{OliveGreen}{\boldsymbol{\times}}$.

But how can this be? Which of the the rules above apply? Am I correct in assuming that current $I$ moving "down" is also the same as a (point) charge $q$ moving down with some velocity $\vec v$?

 

[A.T.]

charge q moving "down"

Positive or negative?

 

[brotherbobby]

Positive or negative?

Positive.

 

[A.T.]

Positive.

Then you have applied the cross product wrongly: v x B points out of the image, not into it. So for a positive charge the force is also out of the image.

 

[brotherbobby]

Then you have applied the cross product wrongly: v x B points out of the image, not into it. So for a positive charge the force is also out of the image.

I don't see how. The RHR (Right Hand Rule) mind you.

The magnetic field is from left to right. Charge is moving down. Hence force points into the page, not out of it

 

[A.T.]

I don't see how.

Then apply the actual formula of the cross product and compute the output vector.

magnetic field is from left to right. Charge is moving down

Note that this is the wrong order of inputs for the cross product. It's v x B not B x v.

 

[PeroK]

I think we had a similar thread a couple of years ago. It might be worth searching for it.

I don't use hand rules, but remember the configuration of the Cartesian unit vectors.

 

[brotherbobby]

Then apply the actual formula of the cross product and compute the output vector.


Note that this is the wrong order of inputs for the cross product. It's v x B not B x v.

Then apply the actual formula of the cross product and compute the output vector.


Note that this is the wrong order of inputs for the cross product. It's v x B not B x v.

Thank you and apologies. A silly error on my part.

Something else if I may. There's an agreement to take the conventional flow of current $I$ from positive to negative. In this problem I raised, namely the directiom of force in a current carrying conductor, the agreement becomes crucial. If you fancy taking current as the flow of electrons $e^-$, you'd find the lorentz force to be indeed into the page (RHR), opposite to the force using the LHR.

 

[A.T.]

the agreement becomes crucial.

What is crucial is that you are using a consistent set of conventions and formulas. But there are several possible sets that would work.

 

[kuruman]

See alternative ways of finding the direction of the vector cross product here.

 

[tech99]

TL;DR: Fleming's left hand rule gives the force experienced by a current carrying conductor to point along the thumb, where the middle finger points along the current and the index along the (magnetic) field. The Lorentz force on a charge moving in a magnetic field is given by Fleming's right hand rule. Taking a moving charge in a straight line to be (crudely) as current, don't the two rules give opposite directions of force on moving charges?

View attachment 369924Situation : I have drawn the image to the right. A current $I$ flows "down" along a wire in a magnetic field $\vec B$ directed to the "right". By Flemings's Left Hand Rule (LHR), the wire should experience force out of the page, as shown by the green bullet $\color{green}\bullet$. However, the current can be crudely approximated by a charge $q$ moving "down" with some velocity $\vec v$. The force on the charge is given by $\vec F=q(\vec v\times\vec B)$.

By using Fleming's Right Hand Rule (RHR) for the same situation, remembering that this is a vector "cross" product, the force on the charge comes out to be into the page, as shown by the mark $\color{OliveGreen}{\boldsymbol{\times}}$.

But how can this be? Which of the the rules above apply? Am I correct in assuming that current $I$ moving "down" is also the same as a (point) charge $q$ moving down with some velocity $\vec v$?

Thank you and apologies. A silly error on my part.

Something else if I may. There's an agreement to take the conventional flow of current $I$ from positive to negative. In this problem I raised, namely the directiom of force in a current carrying conductor, the agreement becomes crucial. If you fancy taking current as the flow of electrons $e^-$, you'd find the lorentz force to be indeed into the page (RHR), opposite to the force using the LHR.

Yes. In the UK we are taught Fleming's Left Hand Rule for motor effect and Fleming's Right Hand Rule for Generator Effect. These rules use the conventional flow direction of electricity, which is from + to -. In the USA I think the Right Hand Rule is used for Motor Effect, and uses the electron flow direction, which is a source of confusion.

 

[A.T.]

I don't use hand rules, but remember the configuration of the Cartesian unit vectors.

It's one thing to remember that configuration, and another thing to rotate it in your head to fit any given scenario. I think people like the hand rules, because they can rotate their hand-axes right in front of the scenario.

But regardless what you use to remember the configuration, you must also pay attention to the order of arguments, because the cross product is not commutative.

 

[kuruman]

It's one thing to remember that configuration, and another thing to rotate it in your head to fit any given scenario. I think people like the hand rules, because they can rotate their hand-axes right in front of the scenario.

I agree. However, too many times I have observed right-handed students during tests use their left hand to figure out the cross product and continue writing instead of putting their pen/pencil down. Prior warnings about this when I introduced the right hand rule went unheeded. Ahh, the dreaded "pressure of the test."

Left-handed students have a clear advantage in this case.

 

[A.T.]

However, too many times I have observed right-handed students during tests use their left hand to figure out the cross product

If you cannot make them stop doing that, then you could teach them doing it the right way with the left hand.

Instead of:
right_thumb x right_index = right_middle

You can do:
left_middle x left_index = left_thumb

 

[kuruman]

If you cannot make them stop doing that, then you could teach them doing it the right way with the left hand.

Instead of:
right_thumb x right_index = right_middle

You can do:
left_middle x left_index = left_thumb

Sure but, to avoid confusion, I would have to ask all left-handed students to leave the room while doing that. I think the best bet is to make sure that every student's left hand knows exactly what their right hand is doing.