Damage of High Voltage Lines
1) The magnetic field of the Earth is on average $B=4*10^{-5} T$. Due to the magnetic field molecules that are charged in the body are deflected. What is the force for a Na+ ion, if the movement of the person is 1 m perpendicular to $B$
2) Some people are concerned regarding the fact that High Voltage Power lines have harmful effects on the human body. Because the electric component of their filed do not penetrate inside the body (water is a dielectric), the harmful effects have to be linked to the magnetic field. For a High Voltage of $500 kV$ at $500 A$ the magnetic field at a distance of 30 m away is $B=3.3*10^-6 T$. In reality the magnetic field is less, because there are multiple lines with opposing currents. If the wires are as in the figure (there is an opposite current at a distance of 4 m), the magnetic field 30 m away is $B=3.3*10^{-7} T$
The frequency of the current is 60 Hz so that it can induce currents in the human body. Say we model the axon (the long part of a nerve cell) as a cylinder of 1 m diameter, having around it Na+ ions. Say also that the magnetic field is parallel to the cylinder.
a) What is the force of the B field on the Na+ ions? What is the effective electric field on the Na+ ions.?
b) The velocity of a molecule inside body is $v=BF$ where B is a proportionality factor (mobility pf molecule). If mobility of Na+ is $B=3.2*10^{-19} m/(Ns)$ what is distance that one ion moves before the magnetic field changes direction? (the size of an ion is $d=10^-10 m$ and it diffuses a distance $D=10^-6 m$ in this time).
1) The Lorentz magnetic force on a Na+ ion is just
$F = q*(v times B)$
$times$ is the cross product of vectors $v$ and $B$
$q =e=1.6*10^{-19} C$ since the ion has only one charge
$v = 1 m/s$ and $B =4*10^{-5} T$
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
$U = -d(phi)/dt =d(B*S)/dt =S*d(B)/dt$
where $phi$ 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.
