The membrane that surrounds a certain type of living cell...

The membrane that surrounds a certain type of living cell has a surface area 8 of $5.0 \times 10^{-9} \mathrm{~m}^{2}$ and a thickness of $1.0 \times 10^{-8} \mathrm{~m}$. Assume that the membrane behaves like a parallel plate capacitor and has a dielectric constant of $5.0 .$ (a) The potential on the outer surface of the membrane is $+60.0 \mathrm{mV}$ greater than that on the inside surface. How much charge resides on the outer surface? (b) If the charge in part (a) is due to $\mathrm{K}^{+}$ ions (charge $+e$ ), how many such ions are present on the outer surface?

The membrane that surrounds a certain type of living cell has a surface area 8
of $5.0 \times 10^{-9} \mathrm{~m}^{2}$ and a thickness of $1.0 \times 10^{-8} \mathrm{~m}$. Assume that the membrane behaves like a parallel plate capacitor and has a dielectric constant of $5.0 .$ (a) The potential on the outer surface of the membrane is $+60.0 \mathrm{mV}$ greater than that on the inside surface. How much charge resides on the outer surface? (b) If the charge in part (a) is due to $\mathrm{K}^{+}$ ions (charge $+e$ ), how many such ions are present on the outer surface?

Key Concepts

Parallel Plate Capacitor

The parallel plate capacitor is a basic electrical circuit component consisting of two conductive plates separated by an insulating material. It is used to store electrical energy in an electric field and its properties can be calculated by considering the area of the plates, the distance between them, and the material in the gap.

Dielectric Constant

The dielectric constant is a dimensionless quantity that describes how much a material can increase the capacitance of a capacitor compared to a vacuum. It represents the material's ability to be polarized by an electric field and is crucial in calculating capacitance when a dielectric medium is present between the plates.

Capacitance Calculation

Capacitance calculation involves using the formula C = (?? * ?_r * A) / d, where ?? is the permittivity of free space, ?_r is the dielectric constant, A is the area of the plates, and d is the separation distance. This formula is essential for determining how much charge a capacitor can store at a given voltage.

Relationship between Charge and Potential

The relationship between charge and potential in a capacitor is described by the equation Q = C * V, where Q is the charge stored, C is the capacitance, and V is the voltage across the capacitor. This relationship allows the calculation of the stored charge when the capacitance and the potential difference are known.

Elementary Charge

The elementary charge is the fundamental charge carried by a single proton (or the magnitude of the charge of an electron) and is denoted by e. It is used to convert total charge into the number of charged particles, such as ions, by dividing the total charge by this basic unit of charge.

Transcript

00:01 In this problem, we are modeling a living cell by a parallel plate capacitor.

00:05 We're given the separation distance, the dialectic constant, which represents the material between the plates, filling between the plates, the area of one of the plates, and we also are told the potential difference between the plates.

00:19 And our goal is to find out two things.

00:24 The amount of charge that's on one of the plates.

00:26 Remember, one plate has plus q, the other has minus q.

00:30 You can think of the electric field as driving off the other positive charges and leaving a negative.

00:38 I think of a traditional circuit.

00:40 So we'll find the charge, q, and then i'm going to be asked how many ions that makes up that charge.

00:49 Now, let's calculate first the charge.

00:52 Q is equal to c times v.

00:55 Formula for c with a dielectric constant, kappa, epsilon 0a over d.

01:01 And then we've got to multiply this by v to get our q.

01:05 And we can put in everything here.

01:07 This is 5 .8 .85 times 10 to minus 12.