Electromagnetism Vectors

[vorcil]

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:

$$\bf{E} = (\hat{x} - 2\hat{y} + 4\hat{z}).$$

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, $$\bf{d} = (2\hat{x} - \hat{y} + 6\hat{z})$$m, Find the work done per unit charge by the electric field?

v) The electric field in iv is applied at the point $$P = (1,2,0)$$ 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 = $$\bf{E.d}$$
Torque on a unit charge, $$\bf{\tau = r x E}$$

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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
$$| E | = sqrt{21}$$

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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

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

(1,-2,4) . (2,-1,6) = 2 + 2 +24 = 28,
I know the units of E = vm^-1 and the displacement 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
$$P = (1,2,0)$$ (which I think is r?)

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

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thank you for checking

 

[lippyka]

my work :)Your work looks correct! To answer your question in part iv, the dot product of E and d is a scalar, so it has no units. The dot product of those two vectors is 28, so the work done per unit charge is 28 joules.