The portion of a nerve cell that conducts signals is called an axon. Many of the electrical properties of axons are governed by ion channels, which are protein molecules that span the axon's cell membrane. When open, each ion channel has a pore that is filled with fluid of low resistivity and connects the interior of the cell electrically to the medium outside the cell. In contrast, the lipid-rich cell membrane in which ion channels reside has very high resistivity. In a simple model of an axon conducting a nerve signal, ions move across the cell membrane through open ion channels, which act as purely resistive elements. Assume that a typical open ion channel spanning an axon's membrane has a resistance of 1×10^ 11Ω. Required:
a. If a typical current density (current per unit cross-sectional area) in the cell membrane is 5 mA/cm2 when the voltage across the membrane (the action potential) is 50 mV, what is the number density of open ion channels in the membrane?
b. If a typical current density (current per unit cross-sectional area) in the cell membrane is 5 when the voltage across the membrane (the action potential) is 50 , what is the number density of open ion channels in the membrane?