Ampere's Law

arutha

The current density inside a long, solid, cylindrical wire of radius a = 10 cm is in the direction of the axis and varies linearly with the radial distance r cm from the axis according to J = J₀ r / a with J₀ =0.5 A/cm² . Find the magnetic field in μT at r = 0.045 m

Ok I know from amperes law that the magnetic field in this case can be derived to be:

B=(mew₀/2*pi)(I*r/R²)

Edit: Wait.. Is that above equation right? That might have been what I did wrong.. When I worked it out I used B= (mew₀*I) / (2*pi*r)

Was that my mistake?



Since it says J = J₀ r / a

And I know current density is I=J*A where A is area

I put those two equations together. Since there is no length of the wire, for Area I just used pi*r^2

J = J₀*r(pi*r^2) / a

Where r=.045m, a=10cm, and J₀ =0.5 A/cm²

Not sure if I have interpreted the question correctly, or if that is the right way to do it.. From there I just put that in as I into the original magnetic field equation, using r on the bottom once again as .045m to get my answer. It was still wrong. Thats my attemp, what did I do wrong?

I converted everything to SI units, to convert A/cm^2 to A/m^2 I multiplied it by 10,000. Is that right?

Anyone know how to get this?
Thanks.

Brad Barker

you are given J = J(r).

in ampere's law, we need to find the current.

so integrate J(r) with respect to area (hint: you'll need to swap dA for dr) to find the current at the value of radius in question.


then apply symmetry arguments to the left hand side of ampere's law, and there ya go.


note: if you want the B-field outside of the wire, you need only integrate up to the actual radius of the wire--J = 0 outside the wire.


drawing pictures helps out, especially when comparing the value of the radius of your amperian loop vs. the radius in question.


if there's any more help needed, just post again.

arutha

Not sure I understand..

Should I get:

B = mew₀*I*r / 2*pi*R²

Where r is the distance to the point in the wire we're looking for, and R is the radius of the whole wire? How does J fit into it?

The part of the question I don't get is this bit:

axis according to J = J₀ r / a

What does that actually mean in terms of the question and how do I get the required information out of it?