Resolving a unit vector from Cylindrical coordinates into Cartesian coordinates

  • #1

Homework Statement


Question 3

(a)A long metal cylinder of radius a has the z-axis as its axis of symmetry.The cylinder carries a steady current of uniform current density J = Jzez. Derive an expression for the magnetic field at distance r from the axis,where r<a. By resolving the cylindrical unit vector eφ along the x-and y-axes, show that the magnetic field at any point P inside the cylinder is

B(x, y, z)= μ2 0 Jz (−yex + xey) ,

where P has Cartesian coordinates (x, y, z), and(x2 + y2) <a2.(15 marks)


Homework Equations



eφ = -sin(theta) + cos (theta)

The Attempt at a Solution



I know that the above equation is relevant but I am not sure how that this resolves into

(−yex + xey).

I would have thought that the only way this works is if sin(90) = 1 and cos (0) = 1 then converting this back to cartesian components would give the required answer.
 

Answers and Replies

  • #2
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Been a while, but does anyone have any further thoughts on this topic?
 
  • #3
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Your equation of eΘ is wrong. It should contain the unit vectors in the x and y directions. Also, geometrically, what is sinΘ and cosΘ in terms of x and y?

Chet
 
  • #4
I am also trying to do this question. Just started this electromagnetism course and no sure where to stat with this question.
 
  • #5
I think I have the derivation part complete and I get

$$B[r] = \frac {\mu_{o} I r}{2\pi a^2}e_{\phi}$$

Then since $$J = \frac{I}{\pi a^2}e_z$$ that substitutes in with J over ez giving me Jz. So then I just need to resolve the r and ephi into cartesian coordinates?
Am I just substituting r for $${x^2 + y^2}^\frac{1}{2}$$ and then use a trigonometric equation for ephi... which I don't know.. I don't really know where to go from chestermillers response. Any tips please?
 
  • #6
Also I'm not sure this question was posted in the right forum, if so could it be moved to the appropriate forum please, introductory or calculus perhaps?
 

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