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Cross product and force on a charge due to applied magnetic field.

  1. Aug 16, 2007 #1
    1. The problem statement, all variables and given/known data

    I hope I am not posting this question at the wrong place, but can someone explain to me the vector product rule, or cross product rule. I don't seem to get the hang of it.

    Why I am asking this is because I want to know how force on a moving charge due to an applied B-field works, how it works in the direction perpendicular to B and v.
    2. Relevant equations

    If A and B are two vectors, then
    A X B = ABsin theta
    the direction of the resultant is perpendcular to the plane in which the two vectors A and B lie.

    For magnetic field,

    F=q (B x v)

    B=applied magnetic field
    v=vel of charge

    3. The attempt at a solution

    We are taught the cross product rule with the help of screw-wrench example. But the main problem here is that in reality, no downward force is acting on the screw: it is just because the screw has helical-ring like structure that, on rotating, it moves downwards. If it was a nail instead of a screw, it would not have moved downwards at all. So I am not satisfied with this explanation of the rule, because it does not explain force acting on a moving charge due to applied field.
    According to me, suppose A and B are two vectors forming an acute angle, where A is the base and B is the moving arm. Then
    i) A.B means: Horizontal component of B, times A.
    For example, F.s (=W) means how much distance is travelled by the object "in the direction" in which force F is applied on it.

    The direction of resultant is the direction of force applied, F.

    ii)A X B means: Vertical component of B, times A.

    The direction of the resultant is always perpendicular to the plane in which A and B lie (How?).

    Regarding magnetism, I know that a moving charge produces its own B-field, and this field is directly proportional to the velocity with which the charge is moving. The applied field only interacts with this magnetic field of the moving charge. Now as far as I know (using the right hand thumb rule), the magnetic field, if traced from N to S, is acting in the same direction of v, that is, in the direction the charge is moving. Now if the charge enters an applied magnetic field B, such that the applied field and v (or magnetic field of charge) is perpendicular to each other, then force acting on charge is max, by the formula:
    F=q Bv sin 90=qBv

    But how do these two magnetic fields interact?
    Edit: I have deleted the diagram I had drawn earlier because it was a wrong one.

    I am only familiar upto high school physics. So advanced maths won't make it easier for me. Please explain in simple terms. I will be grateful.

    Mr V
    Last edited: Aug 16, 2007
  2. jcsd
  3. Aug 16, 2007 #2
    I just remembered that field of a moving charge is in concentric circles. So the applied field B will exert a torque on the electron, which will move it up or down. So I understand the magnetism part. Please just explain the cross product rule.

    Mr V
  4. Aug 16, 2007 #3
    if AxB = C

    Then C is perpendicular to both A and B.
    And, magnitude of C is the area of the llgm, that has sides A and B.

    That's all I can say ><.

    P.S. I know this thing for more than a year, but yet haven't figured out what cross product exactly means ><.
  5. Aug 16, 2007 #4
    What you have written in your P.S. is exactly the problem I have. But even so, thanks for your answer.

    Mr V
  6. Aug 16, 2007 #5
    Cross product is DEFINED as the vector perpendicular to the plane of the two vectors. It is also DEFINED as the area of the parallelogram spanned by the two vectors. It seems like a strange definition, but thats all there is to it.
  7. Aug 17, 2007 #6
    I think I should frame my question differently (it may be a different question altogether, though):

    As far as I know, torque is not a physical thing. I mean that if we are applying a linear force in one direction, we can always oppose it by applying a force in the other direction. But if we want to oppose a torque due to a rotating body, we will have to bring it in contact with a body "rotating" in the opposite direction (or by resisting the rotation, which is also a torque in opposite direction) . Then we can say that the two vectors (i.e. torques) are opposing each other.
    So torques are not "physical" things like forces, which can be added or subtracted by bringing them in contact with each other. We will have to "rotate" a rotating body in the opposite or same direction to subtract or add torques. So we can safely say that 'torque' is a vector provided for measuring rotational force, and that we cannot feel it physically. For example, if we place a top on the palm of our hand and spin it, we won't feel any downward or upward force acting on our hand, which means torque is not a force or anything that can be felt, it is just a means of measurement and calculation. More the rate of change of spin, more torque is said to be possesed by/acting on the top.

    Similarly, linear momentum is also not a "physical" thing.

    But in case of magnetism, there is a physical force experienced by a moving charge in the direction of the cross-product-resultant of (B x v). Can anybody please explain to me how this works out?
    I wrote in my previous post that a torque may be acting on the circular magnetic field of the moving charge, due to the applied magnetic field. But now that I have realized that torque is not a physical thing, that more torque means higher rate of rotation, I am once more clueless as to how this force acting on the moving charge originated.

    Thanks for taking time to read all I have written. If I am mistaken, please correct me. If you have answers, most welcome.

    Mr V
  8. Aug 17, 2007 #7
    Er... anyone there?
  9. Aug 22, 2007 #8
  10. Aug 22, 2007 #9
    How forces arise? Who knows? We experimentally observe that things move, and Newton came up with some rules for that, postulating that force is the time derivative of momentum. Note that just because we think we can "feel" forces says more about evolution than physics. We then observed some charged things moving in a magnetic field, and experimentally determined it to be experiencing a force perpendicular to its motion -- so we use a mathematical gadget to describe it, which is the cross product. Cross products are just maths -- things we use because they fit our physics. It so happens that we use them both in calculating torque and angular-motion-related physics as in calculating the Lorentz force. No deeper connection need be.
  11. Aug 23, 2007 #10
    So, does no one actually know why this force is perpendicular?
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