Potential in human cells. Some cell walls in the human body...

Potential in human cells. Some cell walls in the human body have a layer of negative charge on the inside surface and a layer of positive charge of equal magnitude on the outside surface. Suppose that the charge density on either surface is $\pm 0.50 \times 10^{-3} \mathrm{C} / \mathrm{m}^{2},$ the cell wall is 5.0 $\mathrm{nm}$ thick, and the cell-wall material is air. (a) Find the magnitude of $\vec{E}$ in the wall between the two layers of charge. (b) Find the potential difference between the inside and the outside of the cell. Which is at the higher potential? (c) A typical cell in the human body has a volume of $10^{-16} \mathrm{m}^{3} .$ Estimate the total electric-field energy stored in the wall of a cell of this size. (Hint: Assume that the cell is spherical, and calculate the volume of the cell wall.) (d) In reality, the cell wall is made up, not of air, but of tissue with a dielectric constant of $5.4 .$ Repeat parts (a) and (b) in this case.

Key Concepts

Dielectric Materials

Dielectric materials affect the electric field by reducing the effective field strength within the material. The presence of a dielectric is represented by its dielectric constant (relative permittivity), which modifies the relationship between the electric field and the electric displacement field. This concept is key when the medium between the charged surfaces changes from air (or vacuum) to a tissue with a higher dielectric constant.

Geometric Approximations for Spherical Shells

When estimating physical quantities such as the total electric field energy stored in a cell, it is often useful to model the cell as a simple geometric shape like a sphere. Approximating the cell wall as a thin spherical shell allows one to use geometric formulas to calculate its volume and surface area, which are then integral in determining the total stored energy when combined with the local energy density.

Electric Field Energy Density

This concept pertains to the energy stored in the electric field, quantified by the energy density formula, which is proportional to the square of the electric field strength. By integrating the energy density over the volume in which the field exists, one can determine the total electric field energy. This is particularly relevant when estimating the energy stored in structures like the thin cell wall.

Gauss's Law

Gauss's law states that the total electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of the medium. It is a fundamental principle for calculating electric fields in symmetric situations, such as infinite planes or spheres, and is instrumental in deriving the expressions for the electric field due to uniformly charged layers.

Electric Field from Charged Surfaces

This concept involves calculating the electric field generated by a surface with a uniform charge distribution. When two parallel surfaces carry equal and opposite charge densities, the fields generated by each surface can be superimposed to determine the net field in between. The expression for the field from an infinite charged plane is used here as an idealization, which simplifies the calculation by assuming uniformity and neglecting edge effects.

Electric Potential Difference

The electric potential difference between two points is defined as the work done per unit charge in moving a test charge between those points. In contexts like layered charged surfaces, the potential difference can be determined by integrating the electric field over the distance between the surfaces. This relationship is essential for connecting the electric field to observable voltage differences across a material.

Transcript

00:01 So in this problem, we're discussing cells and cell walls and how it can be basically modeled as a capacitor.

00:08 So we have a charge density of 0 .5 times 10 to the minus 3 kuloms per meter squared.

00:17 So we can call that sigma equals.

00:22 Let's just use the positive side.

00:25 0 .5 times 10 to the minus 3 kouloms per meter squared.

00:32 And let's see what else.

00:37 And the cell wall is 5 nanometers thick.

00:42 So we can say d is equal to 5 times 10 to the minus 9 meters.

00:51 And what else? the cell wall material is air.

00:56 Okay.

00:59 Find the potential, oh, one second, my browser is being funny.

01:18 Okay, we want to find the electric field between and the wall between the layers of charge.

01:26 Okay.

01:30 So the electric field in jump for a capacitor is equal to the charge density divided by epsilon not, so we actually don't even need the distance for this part.

01:44 So we just want to do 0 .5 times 10 to the minus 3.

01:51 And then we want to divide this by epsilon 9, 8 .85 times 10 to the minus 12.

01:58 And for that we get 56 million volts per meter.

02:04 Okay, that's as high as it was in the other bio problem.

02:09 I was checking it quite a few times because i mean it just seems like such a large, electric field.

02:19 But, okay.

02:25 So that's what i got for that one.

02:31 And next, we want to find a potential difference between the inside and the outside of the cell wall and then determine which one's higher.

02:41 So for parallel plate capacitor, e is v over d.

02:49 And so v is e, d, or at least magnitude -wise.

02:57 So we need to take this number and multiply it by this 5 times 10 to the minus 9 distance.

03:07 So i got 0 .28.