Fernando Garcia

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PHYS405 - Electricity & Magnetism I

Spring 2024

Problem 5.6

(a) A phonograph record carries a uniform density of "static electricity" $σ$ . If it rotates at angular velocity $ω$ , what is the surface current density $K$ at a distance $r$ from the center?

(b) A uniformly charged solid sphere, of radius $R$ and total charge $Q$ , is centered at the origin and spinning at a constant angular velocity $ω$ about the $z$ -axis. Find the current density $J$ at any point $(r, θ, ϕ)$ within the sphere.

Part (a)

Recall that

\begin{matrix} (Eq. 5.23, page 219) & K = σ v \end{matrix}

And that for circular motion

v = ω r

K = σ ω r

The direction of $ω$ and $K$ are clear from context, and we may write

K = σ ω r

Part (b)

Recall that

\begin{matrix} (Eq. 5.26, page 219) & J = ρ v \end{matrix}

In this case, we have to account for circular motion at a varying radius: the radius of rotation depends on the radial distance to the origin and the angle $θ$ :

v = ω r \sin (θ) \hat{ϕ}

J = ρ ω r \sin (θ) \hat{ϕ}

Since the (solid) sphere is uniformly charged, we may wish to write the above in terms of $R$ rather than $ρ$ :

\begin{aligned} ρ & = \frac{charge}{volume} \\ = \frac{Q}{\frac{4}{3} π R^{3}} \end{aligned}

Giving:

J = \frac{Q}{\frac{4}{3} π R^{3}} ω r \sin (θ) \hat{ϕ}