- #1
CAF123
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Homework Statement
An insulating disc of radius a has uniform surface charge density σ and is rotating at constant angular velocity w about a perpendicular axis through it's centre.
a)What is the surface current density ##\vec{j}(\underline{r})## at a position ##\underline{r}## relative to the centre?
b)Consider a part of the disc with radii between ##r## and ##r+\delta r##. Find it's contribution ##\delta \underline{B}(z)## to the field on the axis.
Homework Equations
Surface current density ##\vec{j}(\underline{r}) = \sigma \underline{v} = \frac{d \underline{I}}{dr},## where ##dr## is the magnitude of the vector perpendicular to the flow of current.
The Attempt at a Solution
a) ##|\vec{j}(\underline{r})| = \sigma (\omega r )##
b)My method originally was to find the B field contribution from the imaginary loops at radius r and that at radius r + δr and subtract, but that would mean needing the currents contained in these individual loops. I believe this method is incorrect here because in this case I would be ignoring the width of such loops and since we have associated a current density I should be taking the infinitesimal length into account,yes?
So, then ##j = \frac{dI}{δr}##, dI the current in the loop of infinitesimal length (r+δr - r = δr), so dI = σwrδr (This is not a differential equation despite the notation) is the current in that part of the disc between some arbritary r ≤ a - δr and r + δr ≤ a.
I think the definition of surface current density is that it is the current through a length perpendicular to the flow of current. How does this definition hold here? I presume I am not to assume the current is completely in the tangential direction to the loop since I am again given current densities,yes?
The analogous problem of summing up imaginary rings of no finite dimension that constitute the disc is fine. In this case, since there is a current density associated, I think I need to take the width of the rings into account.
Many thanks.