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Homework Help: Electromagnetism Vectors

  1. Jul 19, 2010 #1
    n.b: The following is all math problems, dealing with vectors, I just need someone to check i'm doing the cross products and dot products properly< thank you! :)

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    The electric field exerts a force per unit charge:

    [tex] \bf{E} = (\hat{x} - 2\hat{y} + 4\hat{z}). [/tex]

    i) What is the magnitude of E?
    ii) What angle the E make with the Z axis (polar angle)
    iii) What angle does the projection of the vector on the xy plane make with the x-axis (azimuthal angle)?

    iv) The electric field displaces a charge by, [tex] \bf{d} = (2\hat{x} - \hat{y} + 6\hat{z})[/tex]m, Find the work done per unit charge by the electric field?

    v) The electric field in iv is applied at the point [tex] P = (1,2,0) [/tex] m. What is the torque about the origin on a unit charge placed at P?

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    relevant equations:

    Work done per unit of charge = [tex] \bf{E.d} [/tex]
    Torque on a unit charge, [tex]\bf{\tau = r x E}[/tex]

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    my attempt:
    i) magnitude of E is just the sum of the square of the components,
    x^2 - 2y^2 + 4z^2 = 1^2 -2^2 + 4^2 = 1 +4 + 16 = 21
    [tex] | E | = sqrt{21} [/tex]

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    ii) what angle does it make with the z axis?
    I managed to get 90-arcsin(4/|e|) = 45 degrees

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    iii)What angle does the projection make with the x axis,
    I had, arccos(1/(sqrt(5)) = 27.5 degrees

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    iv)
    the dot product of

    [tex] \bf{E} = (\hat{x} - 2\hat{y} + 4\hat{z}). [/tex]
    and [tex] \bf{d} = (2\hat{x} - \hat{y} + 6\hat{z})[/tex]

    (1,-2,4) . (2,-1,6) = 2 + 2 +24 = 28,
    I know the units of E = vm^-1 and the displacment vector d is m

    what happens when I dot product that? It should be joules, since it is the work done, but i don't understand how to form that from those two unit types

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    v)
    the torque is the cross product of r x E
    [tex] P = (1,2,0) [/tex] (which I think is r?)

    [tex] \bf{E} = (\hat{x} - 2\hat{y} + 4\hat{z}). [/tex]
    2 0 1 2
    -2 4 1 -2
    =
    (8,-4,-4)
    making the torque vector
    [tex] \bf{\tau} = (8\hat{x} -4\hat{y} -4\hat{z} [/tex]

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    thank you for checking
     
  2. jcsd
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