Two closely related questions about current density

[JFuld]

*The problem statement

The second question kinda builds on the first and I am not too confident my solution to the first is correct.

1: A disk has a uniform surface charge density and is rotated at a rate 'w'. Find 'K'.

2: A uniformly charged sphere ('ρ') is spun. Find the current density.

*Relevant equations
K=surface current density = σ*v
J = volume current density = ρ*v


*The attempt at a solution

1: ω= the linear (tangental?) velocity of a point on this disk / radius, so v=wr

then K = ωr*σ , and is oriented in the ∅ direction

2: for a spinning sphere, the velocity of the charge is the same as above except v=0 on the axis of rotation for any r.

so if i let the z axis be the axis of rotation, and let r = r(cos∅+sin∅), and v=wr(cos∅+sin∅)

then J = ρwr(cos∅+sin∅), oriented in ∅ direction


yea that's my attempt at the solution. I think my answers make sense, but most homework problems I am assigned require more than 2 lines of work so it is not unlikley that i am missing something.

thanks in advance for any help

 

[vela]

*The problem statement

The second question kinda builds on the first and I am not too confident my solution to the first is correct.

1: A disk has a uniform surface charge density and is rotated at a rate 'w'. Find 'K'.

2: A uniformly charged sphere ('ρ') is spun. Find the current density.

*Relevant equations
K=surface current density = σ*v
J = volume current density = ρ*v


*The attempt at a solution

1: ω= the linear (tangental?) velocity of a point on this disk / radius, so v=wr

then K = ωr*σ , and is oriented in the ∅ direction

That's correct.

2: for a spinning sphere, the velocity of the charge is the same as above except v=0 on the axis of rotation for any r.

so if i let the z axis be the axis of rotation, and let r = r(cos∅+sin∅), and v=wr(cos∅+sin∅)

then J = ρwr(cos∅+sin∅), oriented in ∅ direction


yea that's my attempt at the solution. I think my answers make sense, but most homework problems I am assigned require more than 2 lines of work so it is not unlikley that i am missing something.

thanks in advance for any help

The speed of a point in the sphere depends on its distance from the axis of rotation. If the axis of rotation is the z-axis, what's the distance of a point (r, θ, φ) from the z-axis?

 

[rude man]

Kind of a strange question.
The current di passing by an arbitrary elemental radial length dr at r, 0 < r < R is
di = σrω dr
so the corresponding elemental current crossing dr is a function of r
and the only meaning for current density must be the current per unit radial length, or
di/dr = σrω which has units of current per unit length.

The total current is i = ∫di = σω∫r dr from 0 to R = σω(R^2)/2 so you could say an average current density is σω(R^2)/2R = σωR/2. The last also has units of current per unit length, the same units as the OP's "K".

But really current density varies from 0 to σωR as we go from the center (r=0) to the edge (r=R) of the disk.

I won't even try for the 2nd part.