1) The Earth has a stationary magnetic field with a magnitude of about 4 1) The
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1) The Earth has a stationary magnetic field with a magnitude of about 4
1) The Earth has a stationary magnetic field with a magnitude of about 4 10?5 T at the surface. When you move charged molecules in your body will be deflected by this field. Calculate the force on a Na+ ion when a person walks at 1 m/s perpendicular to the field. Compare this force to the 3 pN force we calculated for electrostatically interacting molecules (ATP, RNA, myosin) in week 2. 2) There has been some concern that electromagnetic fields from high voltage power lines may cause certain types of cancer. Due to the dielectric properties of water, electric fields do not penetrate living tissue. Therefore, any linkage with cancer is most likely related to magnetic fields. Consider a single high voltage line carrying 500A at 500kV (RMS). An observer standing 30 meters below the line would be subjected to an RMS field of 3.310?6 T. However, the since multiple lines are required to complete the circuit the actual field is much less than this. If the opposing current is on a wire 4 meters above the first wire (fig. 1) the observer experiences a field of only 3.3 10?7 T. The power lines carry an alternating current with a frequency of 60 Hz. The oscillating magnetic field created by this current can induce currents in conductive materials. Axons are long branches of nerve cells that carry signals through your body. We will model the axon as a cylinder 1 m in diameter surrounded by Na+ ions (fig. 2). Take the magnetic field for the pair of wires and assume that it runs parallel to the cylinder. a) Calculate the electromotive force on the ions around the cylinder. What is the effective electric field felt by the ions? b) For molecules in the viscous environment of the cell an intercellular space, velocity is directly related to force according to v = BF where B is the molecular mobility. If the mobility of the sodium ions is 3.2 1019 m/s/N, calculate the distance traveled by the ions before the field reverses direction. Compare this to the ? 10?10 m size of the ion and the ? 10?6 m that the ion would be expected to diffuse in this time.Explanation / Answer
1) The Lorentz magnetic force on a Na+ ion is just
F = q*(v x B)
x is the cross product of vectors v and B
q =e=1.6*10^-19 since the ion has only one charge
v = 1 m/s
B =4*10^-5 T
if B is perpendicular to V then
F =qvB =1.6*10^-19*1*4*10^-5 =6.4*10^-24 N
The force is about F = 6.4 p(pN)
which means it is about 10^-12 smaller than the electrostatic force between interactimg molecules (3pN)
2)
The elecromotice force (said in other words the induced voltage or simply emf) is given by the Faraday law
U = -d(fi)/dt =d(B*S)/dt =S*d(B)/dt
where fi is the total flux through the given area (of the cylinder)
S =pi*d^2/4 =pi*(10^-6)^2/4 =7.85*10^-13 m^2
dt = 1/(60 Hz) = 16.7 ms
d(B) = Bmax-Bmin = 2*B=2*3.9*10^-7 =7.8*10^-7 T
U =7.85*10^-13*7.8*10^-7/16.7*10^-3 =3.67*10^-17 Volts
The electric field felt by the Na+ ions is
E =U/L =U/(pi*d) =3.67*10^-17/(pi*10^-6) =1.168*10^-11 V/m
(Above L is the cylinder perimeter)
2)b)
The force felt by the Na+ ions is
F = e*E = 1.6*10^-19*1.168*10^-11 =1.87*10^-30 N
From text we find that
v = b*F =3.2*10^19*1.87*10^-30 =5.98*10^-11 m/s
The travel time before magnetic field is changing direction is
t = 1/(2F) =1/120 = 8.33 ms
Total distance travelled is
D = t*v =8.33*10^-3*5.98*10^-11 =5*10^-13 m
The distance traveled by the ion is about 200 times smaller than the diameter of the ion itself!
The distance traveled by the ion is also 2,000,000 times smaller than the diffusion distance.
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