Example $7\mathrm{.}4$: $($ i need to answer with details, step by step$)$

A metal disk of radius a rotates with angular velocity $$ about a vertical axis, through a uniform field $B,$ pointing up$.$ A circuit is made by connecting one end of a resistor to the axle and the other end to a sliding contact, which touches the outer edge of the disk $($Fig$.7\mathrm{.}14).$ Find the current in the resistor.

Figure $7\mathrm{.}14$

Solution: The speed of a point on the disk at a distance $s$ from the axis is $v=s,$ so the force per unit charge is $f_{\text{m}\text{a}\text{g}}=vB=sBhat(s).$ The emf is therefore

$E={\displaystyle \underset{0}{\overset{a}{}}}{f}_{\text{m}\text{a}\text{g}}ds=B{\displaystyle \underset{0}{\overset{a}{}}}sds=\frac{B{a}^2}{2}$

and the current is

$I=\frac{E}{R}=\frac{B{a}^2}{2R}$

The trouble with the flux rule is that it assumes the current flows along a well$-$defined path. whereas in this example the current spreads out over the whole disk. It's not even clear what the "flux through the circuit" would mean in this context. Even more tricky is the case of eddy currents. Take a chunk of aluminum $($say$),$ and shake it around in a nonuniform magnetic field. Currents will be generated in the material, and you will feel a kind of "viscous drag"$-$as though you were pulling the block through molasses $($this is the force I called $f_{\text{p}\text{u}\text{l}\text{l}}$ in the discussion of motional emf$).$ Eddy currents are notoriously difficult to calculate, ${?}^5$ but easy