EMF generated in a blood cell by an oscillating magnetic field

In summary, the conversation focused on finding the maximum probability density function (pdf) in relation to sinusoidal waves and derivatives. The participants discussed how the pdf depends on the rate of change of the flux and at what point in the cycle it is maximized. They also considered the equation for the field and the equation for the flux, as well as the angle and the amplitude of the field. The conversation ended with a discussion on the possible impact of the random rotational motion of cells on the rate of change of flux.
  • #1
Jaccobtw
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Homework Statement
Concern is often expressed regarding the potential adverse effects of oscillating electric and magnetic fields on the human body. Consider a typical magnetic field created by mechanical equipment that oscillates at 60Hz and has an amplitude of 1.0x10^−3 T. Determine the maximum emf it can generate in V around a blood cell which has an 8μm diameter.
Relevant Equations
$$\varepsilon = \frac{d\Phi}{dt}$$
$$\Phi = \int_{}^{}B \cdot dA$$
At first I tried plugging everything in with 60Hz in the numerator but that did not work. I was told to think about sinusoidal waves and derivates but I'm not sure how that works. Any ideas? Thanks a lot
 
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  • #2
Jaccobtw said:
think about sinusoidal waves and derivates
Yes. You are asked for the max pdf. As your equations show, the pdf depends on the rate of change of the flux, and that varies during a cycle. At what point in the cycle is it maximised?
 
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  • #3
haruspex said:
Yes. You are asked for the max pdf. As your equations show, the pdf depends on the rate of change of the flux, and that varies during a cycle. At what point in the cycle is it maximised?
The amplitude?
 
  • #4
Jaccobtw said:
The amplitude?
Not the amplitude of the field, no.
Write the expression for ##\phi## as a function of t and apply the equation you quoted to find the pdf as a function of t.
 
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  • #5
haruspex said:
Yes. You are asked for the max pdf. As your equations show, the pdf depends on the rate of change of the flux, and that varies during a cycle. At what point in the cycle is it maximised?
Is it at the inflection point where the slope is greatest?
 
  • #6
Jaccobtw said:
Is it at the inflection point where the slope is greatest?
Yes. Please write the equations I asked for in post #4.
 
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  • #7
haruspex said:
Not the amplitude of the field, no.
Write the expression for ##\phi## as a function of t and apply the equation you quoted to find the pdf as a function of t.
##\phi = (1.0 x 10^{-3}) sin 120\pi t## This equation gives you an amplitude of 1.0 x 10^-3 T and gives 60 cycles per second or 60 Hz. Is it too much? If I take the derivate of this will it lead to the correct answer or am I all mixed up?
 
  • #8
Jaccobtw said:
##\phi = (1.0 x 10^{-3}) sin 120\pi t## This equation gives you an amplitude of 1.0 x 10^-3 T and gives 60 cycles per second or 60 Hz.
This would be the equation for the field ##B(t)## rather than the equation for the flux ##\Phi(t)##.
 
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  • #9
TSny said:
This would be the equation for the field ##B(t)## rather than the equation for the flux ##\Phi(t)##.
so just multiply by the area to get flux, right? does the angle oscillate too or just the magnitude of the field, otherwise I'd include cos##\theta## $$\Phi (t) = (\pi (4 \times 10^{-6})^{2})(1.0 \times 10^{-3}) sin 120\pi t$$
 
  • #10
Jaccobtw said:
so just multiply by the area to get flux, right?
Yes, you need to multiply B by the area.

Jaccobtw said:
does the angle oscillate too or just the magnitude of the field, otherwise I'd include cos##\theta##
The cells will be oriented in all directions. You want the maximum emf. Choose ##\theta## accordingly.
 
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  • #11
TSny said:
Yes, you need to multiply B by the area.The cells will be oriented in all directions. You want the maximum emf. Choose ##\theta## accordingly.
Cool. Then I just take the derivative of that and plug in a max value for t then I get the answer
 
  • #12
Jaccobtw said:
Cool. Then I just take the derivative of that and I get the answer
Basically, yes.

It just occurred to me that a given cell will be randomly changing its orientation with the field direction due to thermal motion etc. Thus, the orientation angle ##\theta## will be time-dependent. So, if you include the ##\cos \theta## in the flux expression, then the time derivative of the flux will include an extra term with a factor of ##\dot \theta##. I don't think you are meant to include this in your analysis, but I wonder if this term might be the dominant contribution to the induced emf. It depends on how fast the cells rotate due to thermal motion, etc.
 
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  • #13
For fun: A very rough way to estimate the rotational rate of spin, ##\omega##, of a blood cell is to use the equipartition of energy theorem. The average kinetic energy associated with a degree of freedom of motion is ##\frac{1}{2} kT##, where ##k## is Boltzmann's constant and ##T## is the absolute temperature. Thus, ##I \omega^2 \approx kT##, where ##I## is the moment of inertia of the cell for rotation about a diameter.

Treating the cell as a disk with a thickness of 2 ##\mu##m and a density approximately that of water, I find ##I \approx 4 \times 10^{-25}## kg m2. With ##T = 300 K##, I find ##\omega \approx 100## rad/s. Or, ##f = \frac{\omega}{2 \pi} \approx 20## Hz. This is in the same ballpark as the 60 Hz applied field. So, the rate of change of flux due to the random rotational motion of the cell is of the same order of magnitude as that due to the changing magnetic field. (Interesting coincidence!)

I could have made some errors here or overlooked something. Corrections welcome!
 
  • #14
TSny said:
the orientation angle ##\theta## will be time-dependent.
Not sure about this. That is as though the cell is a lamina at some angle to the field, but it would be more like spherical. So there will always be a slice of the cell of which the area normal to the field will be changing fastest.
Yes, that gives a term ##\Phi\dot\theta##, but would ##\dot\theta## be anywhere near as much as 60Hz? My impression is that at cellular scales water looks quite viscous, and most Brownian motion would be linear. Probably depends on cell shape; for a perfectly spherical cell there is no obvious reason why impacts should cause it to rotate at all. And most cells would be inhibited from rotating by links to neighbours.
 
  • #15
Yes, good point about the effect of viscosity. And applying the equipartition of energy theorem to a degree of freedom of rotation of the entire blood cell is probably silly. The cell is huge compared to molecular dimensions. Regarding the disk-like shape of the cells, see here.

This video shows blood cells in motion. I don't know if the video is slowed down, but it appears there's not a lot of rotation going on.
 
  • #16
Something else, an electric or magnetic field that oscillates w.r.t time variable will also oscillate w.r.t spatial variable, that is it will be a wave, however at 60hz it turns out that the wavelength is ##\frac{c}{f}=\frac{3\times 10^8}{60}=5\times 10^6m=5000km##, so its practically not spatially varying within the diameter of a bloodcell of ##8\times 10^{-6}m##.
 

1. What is an oscillating magnetic field?

An oscillating magnetic field is a type of electromagnetic field that is constantly changing direction and amplitude over time. It is typically generated by an alternating current (AC) source.

2. How does an oscillating magnetic field affect a blood cell?

When a blood cell is exposed to an oscillating magnetic field, it can induce an electrical current within the cell due to its conductive properties. This can cause changes in the cell's membrane potential and ion concentrations, potentially leading to cellular damage.

3. What is the relationship between EMF and blood cell damage?

Studies have shown that exposure to high levels of EMF, particularly from an oscillating magnetic field, can lead to damage in blood cells. This can manifest as changes in cell morphology, decreased cell viability, and increased oxidative stress.

4. Can EMF generated by an oscillating magnetic field be harmful to human health?

While the effects of EMF on human health are still being studied, there is evidence to suggest that exposure to high levels of EMF, including that generated by an oscillating magnetic field, can have negative impacts on cellular function and potentially lead to health issues.

5. How can we protect ourselves from EMF generated by an oscillating magnetic field?

To minimize exposure to EMF, it is recommended to limit the use of electronic devices, especially those that emit high levels of EMF. Additionally, using protective shielding materials and keeping a safe distance from EMF sources can also help reduce exposure.

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