Linear vs Convergent Diffusion

Show that the ratio of diffusion-limited currents between a spherical electrode of radius and a planar electrode of the same area is given by  

The diffusion-limited current to a planar electrode is given by the Cottrell equation  

The physical significance of the result is as follows. For short times where TvDt ( fg, the size of the spherical electrode greatly exceeds that of the diffusion layer, and hence the former experiences effectively planar diffusion (Fig. 5.2). 

On the other hand, when irDt r, the current at the spherical electrode approaches steady state due to convergent diffusion. The size of the diffusion layer greatly exceeds that of the electrode. Note that the ratio of currents tends to infinity as the time gets longer, since whereas convergent diffusion sustains a steady-state current at the spherical electrode, the current at the planar electrode, which necessarily experiences only planar diffusion, collapses to zero. 

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