A model of a red blood cell portrays the cell as a spherical...

A model of a red blood cell portrays the cell as a spherical capacitor, a positively charged liquid sphere of surface area A separated from the surrounding negatively charged fluid by a membrane of thickness t. Tiny electrodes introduced into the interior of the cell show a potential difference of 100. mV across the membrane. The membrane’s thickness is estimated to be 100 . nm and has a dielectric constant of $5.00 .$ (a) If an average red blood cell has a mass of $1.00 \times 10^{-12} \mathrm{kg}$ , estimate the volume of the cell and thus find its surface area. The density of blood is $1.10 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$ (b) Estimate the capacitance of the cell by assuming the membrane surfaces act as parallel plates. (c) Calculate the charge on the surface of the membrane. How many electronic charges does the surface charge represent?

Key Concepts

Electric Potential Difference

The electric potential difference, or voltage, is a measure of the work done per unit charge to move a charge between two points in an electric field. It plays a central role in capacitor behavior as it directly relates to the stored charge through the equation Q = CV. Understanding this relationship is critical in setups where measuring or applying a specific voltage is necessary to determine other electrical properties.

Charge Quantization

Charge quantization is the principle that electric charge exists in discrete amounts, with the elementary charge (the magnitude of charge on an electron or proton) serving as the smallest unit. This concept is important when estimating the total number of elementary charges corresponding to a given total charge, highlighting the discrete nature of electrical charge in microscopic systems.

Spherical Geometry in Physical Models

Spherical geometry is frequently used in physics to approximate the shape of objects when their dimensions allow for such simplification. Important geometric relationships, such as the volume V = (4/3)?r³ and surface area A = 4?r² for a sphere, are used to relate the physical properties of spherical objects. These relationships enable the estimation of key parameters that feed into further calculations in models involving electrostatics or other fields.

Dielectric Materials

Dielectric materials are insulating substances that become polarized when an electric field is applied, which reduces the effective field within the material. The dielectric constant (or relative permittivity) quantifies this property, affecting the capacitance of a capacitor. When a dielectric is inserted between the plates or surfaces of a capacitor, the capacitance increases by a factor equal to the dielectric constant, making this concept crucial for designing and analyzing capacitors with non-vacuum media.

Capacitance

Capacitance is a measure of a system's ability to store electric charge per unit potential difference. In many problems, the relationship C = Q/V is used where C is the capacitance, Q is the stored charge, and V is the potential difference. This concept is essential in understanding energy storage in electrical systems and is applied in a variety of geometries, from parallel plate capacitors to more complex shapes such as spherical capacitors.

Parallel Plate Approximation

The parallel plate capacitor model is a simplified way to estimate capacitance by assuming two conductive surfaces separated by a dielectric medium, with uniform electric fields between them. While many real-world capacitors may not have a perfect parallel configuration, this approximation (using C = ???A/t, where A is the area and t is the separation) provides a useful and often reasonably accurate method to calculate capacitance, especially when the separation is much smaller than the dimensions of the plates.

Transcript

00:01 Here the goal is to figure out the amount of charge that a cell membrane has stuck to its surfaces.

00:11 So we're going to treat the cell membrane as a capacitor with two plates of equal area.

00:17 The outer and inner walls of the membrane are serving as the plates that are building up a positive charge on the inside, a negative charge on the outside.

00:30 So a reminder that we've determined capacitance from the area of a plate times an electrical constant divided by the spacing between the plates.

00:46 And let's see, we have the electrical constant, and we are given the spacing.

00:54 But what we don't know is the area.

00:59 So if we knew the radius of the cell, we could find the area.

01:03 And so we're kind of kind of go through that route to find the area.

01:10 By first of all, finding the volume is equal to the mass of the cell divided by its density.

01:23 And really, we're finding the mass and density of the interior.

01:31 But using the numbers given, we come up with 9 .09 times 10 to the mind.

01:40 16 cubic meters.

01:48 So just using the mass and the density given.

01:52 And now we want to find the radius.

01:55 So the volume of a sphere is four -thirds pi are cubed.

02:04 And inverting that to find the radius, take a cube root.