Force due to magnetic field

1. Jan 4, 2016

Amr Elsayed

Hello all,
I know that the force due to magnetic field on a wire containing current = the length of that wire* the current intensity x magnetic field upon that wire, but I fail to understand where the "the length of that wire* the current intensity" comes from or why it's even there.
Sometimes I see it replaced by Q*V. You may say this actually equals Q* d/t which is the same as I*L, but this doesn't appear to be the case because Q/t here is meant to be the charge passing through a certain length " not equal to zero" in a certain amount of time. The problem stems from the definition of current intensity. it's said to be the charge passing through a cross section per second, and it appears to be just that sample. However, cross section without defining a thickness makes no sense to me.
I just need some explanation for why Q*V is the same as I*L and why do we have to cross product them with magnetic field. A fulfilled physical definition of force due to magnetic field might help.
I hope you understand my problem
Regards

2. Jan 4, 2016

gleem

What is the level of your knowledge of physics?

3. Jan 4, 2016

Amr Elsayed

High school stuff, i'm still a second grader.

4. Jan 4, 2016

gleem

OK lets try this , a little history first. Andre Ampere was the first to report the observation that magnetic fields create a force on a wire carrying current. He deducted the relationship of the force between two wires carrying currents from experiments. It was already known that a current in a conductor produces a magnetic field. Of course they also knew that fixed electric charges exerted forces on one another. But an electric current has no net charge so this force was quite different and comes into play when charges move as with an electric current. Also at this time the electron was not yet discovered.

Now a current has not only a magnitude but to completely specify it you must include its direction. It is a vector quantity. Are you OK with this? So we now have an interaction between a magnetic field which is also a vector quantity and a current. So one would expect that there ought to be a dependence on their relative orientations.

In fact His discovery led to the relationship of the force between two conductor where the problem is reduced to stating the relationship between pieces of currents called a current element , short lengths of conductor carrying a current and pointing in a specific direction at that point in each conductor. Since a conductor with a current can be of any shape, and size He need to express this force in a relationship showing how a very short length of a conductor with a certain current and direction at an arbitrary point affects a similar current element at a different point in space associated with the other the other conductor. In fact He also showed that the force depended on the sine of the angle ( you know what that is I assume) between the two current elements direction. Using The Calculus as His mathematical tool He was able to calculate the force between any two arbitrary electrical circuits.

His law can be put into a form that relates the force on an arbitrary circuit element and any magnetic field not necessarily from another current carrying wire.So for the simple case of a straight wire in a homogeneous ie constant magnetic field He gets that the force depends on the current in the wire, the length of the wire, the strength of the magnetic field and its direction by way of the sine of the angle between the magnetic field and the wire.

Now I think it is pretty intuitive that the larger the field the stronger the force. Also since in this case the wire experiences the same force at each point that the total force must be the sum of the force on each current elements therefore on its length. And since the conductor has same direction at every point as does the field then their respective orientation is constant and the force must depend on the sine of that relative angle.

So the result is a conclusion of the observations made in his original experiment.

Perhaps someone else can enter his/her explanation. Please let me know where I can elaborate if I have not properly addressed you request.

5. Jan 5, 2016

Amr Elsayed

Thank you first, this allegedly gives a good intuition about why those variables are considered in our relation. Even though, I assume my problem is mathematical.
I wanted to know exactly why force due to magnetic field is given by this exact relation, till no I don't have a fulfilled definition of force due to magnetic field. I think it will greatly help because it will relate with the relation for sure. I wonder if it means the sum of the magnetic field exerted upon all charges.besides, my geometrical understanding of cross product doesn't explain why cross product is present in the relation.
Regarding mathematics, I know basic algebra and geometry besides limits and differentiation and I also studied vectors in mechanics before.
Thank you
Regards

6. Jan 5, 2016

Amr Elsayed

BTW you might get deeper in any math that the question is concerned with as long as it's in my understanding area of math. This is actually why I studied some calculus out of the curriculum.

7. Jan 5, 2016

gleem

It is found that a magnetic field that points in the direction of a current exerts no force on the current as determined by observation As the angle between the current and the field increases so does the force. This suggests that the force is due only to that component of the magnetic field that is perpendicular to the current. I will assume you are familiar with elementary vector algebra and the resolution of a vector into mutually perpendicular directions.so the relationship between the current and the field could be expressed by the equation

F = k*I*Bperp

where Bperp is the component of the magnetic field at right angle to the current.

The field is a vector quantity and as such can be written as the sum of two vectors at a right angle to one another. In fact this is an infinite set of vectors until you choose a direction of interest whereupon you can then write the specific vectors which add to form the the actual field vector The magnitudes of these vectors are BcosΘ and BsinΘ for the parallel and perpendicular component of B respectively and Theta is the angle between the actual magnetic field and your chosen direction which in this case is the direction of the current element.

Since the force exerted by the parallel component of B is zero

You get F = k⋅IB⋅sinΘ or in vector notation F =k⋅I×B where the bold letters stand for vector quantities.

8. Jan 5, 2016

Amr Elsayed

Yup, I am familiar with what you did when considering both components of the vector. But still I can't find the answer to my question. I just need to know what is " force due to magnetic field", a physical definition will be okay.

9. Jan 5, 2016

gleem

The force due to a magnetic field is the force that acts on an electric charge that is moving relative to the magnetic field and whose direction is perpendicular to its motion thus resulting in a change in direction of the charge with no affect on its speed.. That's a definition but you know this.

10. Jan 6, 2016

Amr Elsayed

Oh, then what do we mean when we say force due to magnetic field upon a wire ? do we mean it to be upon all the charges ? why including the current intensity or the velocity then ??

11. Jan 6, 2016

gleem

Note that I said that it was the force on "moving charges" which of course constitutes a current. A charge at rest with respect to the magnetic field experiences no force.

Consider in a wire an amount of charge Q spread out in a length of wire L so you have Q/L coulombs per meter. If the charges Q moves with a velocity V you have a flow of charge. The quantity (Q/L)V is equal to coulombs per second therefore is a current.

A wire at rest with no current experiences no force but when the charges move to produce a current the wire experiences a force at every point depending on the amount of charge, the speed and the field strength at that point and of course the angle between the current direction and the field direction.