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I Why do physicists show the magnetic field perpendicular to F

  1. Nov 26, 2016 #1
    I mean physicists when they have to create a field they draw it with the same direction of the force, (gravitational, electric for example) i know that the magnetic field acts only on moving charges and i also know that the force is a relativistic effect of lorentz contraction, but why creating a field and making it perpendicular to a force? why not representing the magnetic field vector perpendicular to the force?

    Trying to answer myself, please tell me if i got it or if it's wrong:

    so, we are trying to find a vectorial field, we know from experiments that the force created by this field is proportional to the velocity of entry of a charge and to the strenght of the field, the force is a vector. the field is a vector, the velocity is a vector, the only way we can get a vector F from v and B also vectors so that F is proportional to both is by cross product, this implies that the F is perpendicular to B so we make a vectorial field perpendicular to the force. basically we see that the force changes direction with the velocity and that has to be because of another vector acting on the velocity

    is this correct?
     
  2. jcsd
  3. Nov 26, 2016 #2
    B is actually called magnetic inductance and I think it's for a good reason.

    The electric field at some position will vary with charge variation like this :
    dE = dq r / (4πε r3) ,
    with "bold" r as position vector.
    And the force acting on another particle is written as
    F = q E = q1 q2 r / (4πε r3)

    From Biot Savart we have
    dB = μ I dl x r / (4π r3)
    = μ (q/dt) dl x r / (4πr3)
    = μ q v x r / (4π r3),
    with vectors l, r and v.
    So, the magnetic field induction is produced by a charge moving (or a current flow).

    The Lorentz force is written as
    F = q v x B
    which can be detailed further more as
    F = q1 v1 x (μ q2 v2 x r / (4π r3))
    with r being the distance between q1 and q2.

    So inside the Lorentz force we have 2 curls :
    - first curl with the affected particle's velocity and the induction
    - second curl inside induction, from a charge's movement and the distance from the charge.
    v1 x (v2 x r) (PS: curl is not commutative)

    So overall force is
    (1) F = m a = m dv/dt

    (2) F = q E + q v x B
    F = q1 q2 r / (4πε r3) + q1 v1 x (μ q2 v2 x r / (4π r3))

    From (1) and (2) we get that

    m dv1/dt = q1 q2 r / (4πε r3) + q1 v1 x (μ q2 v2 x r / (4π r3))

    (To cover relativistic effects one will use : dpα / dτ = q Fαβ Uβ )

    Now if there were magnetic charges we would call them g (or magnetic "monopoles").
    Dirac developed the theory for magnetic charges (symmetric electromagnetic field theory) and it's like this :

    g = ∫∫∫volume ρm dV = ∫∫area B dS

    with ρm as magnetic charge density.
    ( the elementary magnetic charge would be g = ħ/(2eμ) = 2.6 10-10 [Am] )

    B = μ g r / (4π r3)

    ∇ x H = J + ∂D/∂t
    ∇ x E = -Jm - ∂B/∂t
    ∇D = ρ
    ∇B = ρm

    And a force acting on another magnetic charge g would be defined as :

    F = g B
    which is similar to F = q E.

    So, between charges of same type - electric vs electric, magnetic vs magnetic -we have "direct" force (no curl), the force has the same direction with the field.
    In short the answer to your question is :
    Since magnetic induction is generated by moving charges it acts only on moving charges inflicting same type of "action" (or same kind of "response").

    This is the theoretical layer of classical electromagnetism and I might have gotten more deeper than you asked :smile:
    If there's any mistake i think someone else will cover for it.

    PS: one more thing.
    (1) Electric field changes and magnetic field changes propagate with the speed of light so F = q v x B will happen when the field (change) arrives at the destination charge (aka [time of arrival] = [distance between charges] / [speed of light] ).
    (2) the spin is a charge "moving" or better said "rotating". So the "spinning" of charge will be affected by the magnetic inductance.
     
  4. Nov 26, 2016 #3

    PeroK

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    In which direction should the magnetic force field point? Given that charged particles will move in very different directions depending on the direction of their velocity.
     
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