EMF generated in a blood cell by an oscillating magnetic field

[Jaccobtw]

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

 

[haruspex]

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?

 

[Jaccobtw]

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?

 

[haruspex]

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.

 

[Jaccobtw]

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?

 

[haruspex]

Is it at the inflection point where the slope is greatest?

Yes. Please write the equations I asked for in post #4.

 

[Jaccobtw]

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?

 

[TSny]

$\phi = (1.0 x 10^{-3}) sin 120\pi t$ 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 $\Phi(t)$.

 

[Jaccobtw]

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$\theta$ $$\Phi (t) = (\pi (4 \times 10^{-6})^{2})(1.0 \times 10^{-3}) sin 120\pi t$$

 

[TSny]

so just multiply by the area to get flux, right?

Yes, you need to multiply B by the area.

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.

 

[Jaccobtw]

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

 

[TSny]

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.

 

[TSny]

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!

 

[haruspex]

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.

 

[TSny]

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.

 

[Delta2]

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 $8\times 10^{-6}m$.