# Resolving a unit vector from Cylindrical coordinates into Cartesian coordinates

## 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.

Been a while, but does anyone have any further thoughts on this topic?

Chestermiller
Mentor
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

I am also trying to do this question. Just started this electromagnetism course and no sure where to stat with this question.

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?

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?