Consider a spherical biological cell of outer radius, Router = 5.0 times 10-6 m.

Question

Consider a spherical biological cell of outer radius, Router = 5.0 times 10-6 m. The membrane thickness of cell is 5.0 times 10-9 m. For this cell, the potential difference across the cell membrane (Vin -Vout) is -80 mV and the dielectric constant for the cell is k = 7.0. Note: The cell membrane has capacitive properties no different than the capacitors used in the lab. What is the capacitance of the cell? How much charge is stored on each face of the membrane? How much potential energy is stored with in the cell membrane?

Explanation / Answer

(a)   capacitence of a spherical capacitor                C   = k ( 40 )   ( Rinner Router / (Router  - Rinner  )   ..........(1)           Here k = dielectric constant = 7.0           40 =   1/ ( 9*109 )  C2 / N.m2         Router   =   5.0*10-6 m ,   Rinner    =  Router    - thcknessof the shell                    Rinner     =   4.995 *10-6 m        plug all values in (1), we get                C = 3.885 *10-12 F   =   3.885 pF (b)  magnitude of   Charge   q   = CV   =   ( 3.885 *10-12 F )( 80*10-3 V )                  q =  0.311  *10-12 C =   0.311 pC    The charge on the inner suface of the sphere    =   +q   = +   0.311  *10-12 C   The charge on the outer surface of the capacitor   = - q =   -0.311  *10-12 C   (c) The potential energy            U =   (1/2) CV2      =   (1/2)(   3.885 *10-12 F ) ( 80*10-3 V ) 2                                              =   1.24*10-14 J                                              =   1.24*10-14 J

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