Lorrentz force law & vector notation

[Roodles01]

Homework Statement

A current element is 3.0mm long, centerd on the origin of cartesian coordinates, and carries a current of 2.5A in the direction e_z.
What field does it produce at the point (3.0, 0, 4.0)

Homework Equations

This is obviously a Lorentz force law question
so . . . ∂B = (I ∂I (r-r₀)) / I r-r₀ I^3

where B = magnetic field
I - current
r - point of origin
r0 - point being considered

The Attempt at a Solution

∂B = 4∏x10^-7 / 4∏ x 2.5A x 3.0x10^-3 )(e_z(3e_x + 4e_z)) /5^3

cancelling 4∏'s
∂B = 10^-7 x 2.5A x 3.0x10^-3 (e_z(3e_x + 4e_z)) /5^3

The number bit is easy, but the problem I have is the vector notation.

Please, how do I do the (e_z(3e_x + 4e_z)) /5^3 bit?

I have the final answer as
1.8x10^-11 T e_y
[

 

[milesyoung]

I might be missing something, but shouldn't you be considering the Biot–Savart law instead?

It seems to me like you're being asked to determine the magnetic field produced by a current and not the force on a point charge in an EM field.

Edit: I see hints of the Biot–Savart law in your expressions, so I guess it was just a typo.

If you start out with the differential:
$$d\mathbf{B} = \frac{u_0}{4\pi}\frac{I d\mathbf{s}\times \mathbf{\hat{r}}}{r^2}$$
where $d\mathbf{s}$ is directed along the current and $\mathbf{\hat{r}}$ is a unit vector towards the point you're considering.

Then since $d\mathbf{s}\times \mathbf{\hat{r}}$ is always directed along $\mathbf{e}_y$, you have:
$$d\mathbf{B} = \frac{u_0}{4\pi}\frac{I |d\mathbf{s}\times \mathbf{\hat{r}}|}{r^2}\mathbf{e}_y$$
The problem is then to find expressions for $d\mathbf{s}$ and $r$ in terms of a common variable you can use to integrate over the whole of the current element.

Any help?