PHYS2x12 Electromagnetic Properties of Materials Assignment 2022
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PHYS2x12 Electromagnetic Properties of Materials Assignment 2022
October 26, 2022
Due date: Sunday October 30
In each of the questions below marks will be awarded for clear explanations, not just giving the correct answer. In this assignment we will consider some of the electromagnetic effects that are relevant to biological systems.
1. 15 marks Consider a model of a spherical cell of radius r . The interior of the cell we will model as a conducting sphere of radius r _ a. This is separated by a lipid layer of thickness a and dielectric constant K from the exterior of the cell which we will also model as a perfect conductor. The interior of the cell is negatively charged.
With these assumptions use Gauss’s law to determine the capacitance of the cell.
Follow the following steps, that correspond to what we did in class for the parallel plate capacitor
(a) Suppose there is a charge _Q on the cell. Argue that the electric field must be spherically symmetric. (b) Use Gauss’s law for a material with dielectric constant K to find the electric field in the cell membrane
(c) Use the definition of electric potential to find the potential difference across the cell membrane
(d) Use the definition of capacitance to find the capcitance from the electric potential
2. 10 marks Assume that for our spherical cell we have r = 5 µ m, a = 0.005 µm, K = 11, and V = _100
mV. These are all values that are somewhat typical of at least some real cells.
(a) Find the capacitance C for our spherical cell
(b) Compare this value to the capacitance of a parallel plate capacitor with plate separation a and area equal to the surface area of the cell. Why are these values so similar?
(c) Find the amount of charge Q stored in the cell
(d) Find the charge density of free charge on the surface of the cell (e) Find the charge density of bound charge inside the cell membrane.
(f) Draw a diagram of the cell membrane indicating the electric field, the polarization density, and the bound and free charge.
3. 10 marks The charge in a cell results from the fact that the material inside the cell has an excess of K+ ions relative the material outside the cell. The cell has ion channels that allow the K+ ions to diffuse out of the cell. This leads to a build-up of positive charge outside the cell and an electric field across the cell membrane. Due to this electric field the potassium ions will tend to flow back into the cell with a current density equal given by Ohm’s law
J = σE
where σ is the conductivity of the K+ ions through the ion channels. Equilibrium is reached when the rate of diffusion out of the cell equals the rate at which the electric field brings ions back into it.
Diffusion is governed by Fick’s law. Since the density of K+ ions N (r) is spherically symmetric Fick’s law states that the current density due to diffusion out of the cell is radial and spherically symmetric and given by the equation
dN
Jr,diff = _qD
dr
where q is the charge of the ions and D is the diffusion constant. Statistical mechanics and plasma physics
shows that the diffusion constant is
σkBT
D =
q2N
(a) Use the fact that the electric field is radially and spherically symmetric to find a differential equation for the electric potential in terms of the density of potassium ions when the current due to Ohm’s law and the current due to diffusion are equal and opposite.
(b) Solve the differential equation to show that the potential difference across the cell membrane is
kBT
q
where Nin and Nout are the density of potassium ions inside and outside the cell respectively. (The logarithm is base e.)
(c) If Nin = 120 mmol/L and Nout = 5 mmol/L find the potential difference at body temperature (T = 310
K) and compare to the voltage we assumed in the previous question.
4. 15 marks Many fish and other aquatic animals can detect electric fields. A famous example is sharks whose sensitivity to electric fields is sufficient to detect the EMF induced in their body as they move through the Earth’s magnetic field.
The electrically sensitive part of the shark is known as the ampullary organ. We can think of the ampullary organ as a highly conductive rod of length l surrounded by highly resistive parts of the body of the shark. An electric field in the ampullary organ causes a current to flow along the rod, out through the resistive body, and back to the start of the rod through the conductive ocean water. We will use R to represent the total resistance of this circuit.
At the end of the ampullary organ is a receptor that actually generates the electric signal that travels to the shark’s brain. We can think of the receptor as a resistor of resistance Rr which develops a voltage across it due to the current flowing. This voltage ultimately triggers neurons to carry a message to the brain. (The total resistance R includes the contribution of the receptor.)
The ampullary organ of a shark can detect electric fields between 5 nV/cm and about 1 µV/cm.
We will suppose that the electric field in the ampullary organ is generated by the EMF induced as a conductor is moved through the Earth’s magnetic field.
(a) Suppose that the shark has an ampullary organ oriented vertically. Find an expression for the potential difference between each end of the rod for a shark moving with velocity v→ in a magnetic field 
.
(b) Draw an effective circuit diagram for the ampullary organ, showing the EMF due to the shark’s motion, a resistor for the receptor, and a resistor representing the rest of the shark’s body.
(c) Find an expression for the voltage across the receptor.
(d) For a shark swimming in Sydney harbour at 0.5 m/s what is the size of this EMF per centimetre of rod in the ampullary organ.
(e) Is this electric field in the range that can be detected by a shark?
(f) Suppose that Rr/R ~ 0.5 and rod is 10 cm long, what must be the minimum and maximum detectable voltages across the receptor? How does this compare to the typical voltage across a cell wall?
(g) As the shark moves it swings its head left and right relative to its body at angular frequency ω . The angle between the shark’s head and its body is θ = α sin(ωt). The body of the shark continues to move with constant velocity v→ the constant velocity of the shark’s body. If α is small, find an expression for the EMF as a function of time.
(h) The horizontal component of the magnetic field 
points north. Find approximate expressions for the time dependence of the signal seen by the shark when it is moving in a horizontal direction: north, south, east, and west.
(i) Comment on which directions result in the largest time dependence of the EMF. Is it possible to tell north from south or east from west? Think about the fact that the shark knows both which direction its head is moving and the sign of V .
(j) Why do you think that many aquatic vertebrates can sense electric fields, but very few terrestial vertebrates can?
2022-11-16