Geomagnetic Cushioning (19 July 2024)

I’ve agonised over the experience of dropping valued electronic gadgets to the ground. There came a point when I imagined how alleviating it would be if we could add functionality to these devices such that when you drop one it wouldn’t fall straight to smash on the ground but rather cushion itself better than phone cases do. 

I couldn’t help feeling inclined to explore a scheme where the device would detect it was falling and use it’s electrical energy to transiently enact this protection. Being me, I gravitated towards looking for some property of the earth that could interact with this momentary state of the device, say, a phone. So I thought of the earth’s magnetic field. If we went down this alley we might say of the concept, “We’ve conjured up a form of ‘geomagnetic cushioning’.”

Entertaining the idea further I thought, from electrodynamics we’re  cognisant of a principle known as the Lorentz force law (which I've mentioned in prior articles) that informs us a charged particle in motion in a magnetic field experiences a force at right angles to the direction of the motion of the particle and that of the magnetic field and proportional in magnitude to the product of the velocity of the particle with the magnetic flux density. Formally,

\begin{array}{c}\textbf{F} = \mathrm{q} \textbf{U} \times \textbf{B} \ \ \ \mathrm{Equation (1)} \end{array}So if the falling phone would use an inclinometer and accelerometer embedded in it to detect the falling event and instantaneously effect an electric charge on capacitor plates at it’s front and/or rear faces then it wouldn’t fall vertically but glide to the ground (due to the geomagnetic field) as if it had let  out a parachute. This is illustrated in Fig. 2. 

The configuration of the plates in a phone in my proposed scheme would roughly be as in Fig. 1 below in which the right or left view of the phone is shown with both capacitors charged in the falling event. Perhaps the device could be smart enough to charge only the face it currently predicts will hit the ground first based on instantaneous data from the above mentioned sensors.

Fig. 1 Side-view of Phone with Capacitors Activated

In this item I’d like to investigate the values we’d need to attain for parameters in our design of this subsystem to make the cushioning achievable. 

First, to find the time, t_G, required for a body in free fall (i.e. neglecting air resistance) to travel a vertical distance, s, we may resort to a high school physics formula;

\begin{array}{rcl} s & = & \mathrm{ut} + \frac{1}{2}\mathrm{at}^{2}\\ & = & \frac{1}{2}\mathrm{at}^{2}\\ \therefore \ \mathrm{t}_{G} & = &\sqrt{\frac{\mathrm{2s}}{a}} \\ &= & \sqrt{\frac{\mathrm{2s}}{g}} \ \ \mathrm{Equation (2)} \end{array}We then consider the influence of the earth’s magnetic field (geomagnetic field) on the phone's trajectory as illustrated in Fig. 2. 

In this discussion the magnetic flux density we consider is the component B_y of the geomagnetic field with reference to Fig. 2. 

Fig. 2 Cartesian Plane Representation Of The Trajectories

If we can find an expression for the vertical distance traveled, s, at which full deflection is achieved by the Lorentz force we can then tune parameters in our design to meet their required values for s.

To that end, in the following we first find an expression for the time, t_L, that it takes for a falling charged body to be completely deflected from free fall by means of the Lorentz force due to the geomagnetic field:$$\begin{array}{l}Let \ the \ body's \ peak \ velocity \ in \ the \ z \\ direction \ be \ \textbf{u}_{z0} = u_{z0}\textbf{a}_{z}. \\Since \ kinetic \ energy \ is \ conserved \\ under \ the \ Lorentz \ force \ the \ magnitude \\ of \ the \ peak \ velocity \ in \ the \ x \ direction, \\ {v}_{x0} = u_{z0} \ but \ \mathrm{v}_{x0} \ = \ A_{x}\mathrm{t}_{L} \ \mathrm{Equation (3)} \\ where \ t_{L} \ is \ the \ time \ to \ full \ deflection \\ and \ A_{x} \ is \ acceleration \ in \ the \ x \ direction. \\ The \ Lorentz \ force \ acting \ on \ the \ body \ , \\\textbf{F}_{x} = Q\textbf{u} \times \textbf{B}_{y} = mA_{x}\textbf{a}_x \\ so \\ \begin{array}{rrlcl} \ & & A_{x}\textbf{a}_{x} & = & \frac{Q\textbf{u} \times \textbf{B}_{y}}{m} \\ & & & = & \frac{Qu_{z0}\textbf{a}_{z}\times \textbf{B}_{y}}{m} \\ & & & = & \frac{Qv_{x0}\textbf{a}_{z}\times \textbf{B}_{y}}{m} \end{array} \\ yet \ from \ Equation \ (3); \\ \begin{array}{rrlcl} \ & & t_{L} & = & \frac{v_{x0}}{A_x} \\& & & = &\frac{v_{x0}m\textbf{a}_x}{Qv_{x0}\textbf{a}_{z}\times \textbf{B}_{y}} \\ & & & = & \frac{m}{Q\mathrm{{B}_{y}}} \ \mathrm{Equation (4)} \end{array} \end{array}$$So to deflect the phone within a period t_G we relate t_L to t_G from equation (2):

\begin{array}{rcl} \mathrm {t_{G}} & \geqslant &\mathrm {t_{L}}\\ \sqrt{\frac{\mathrm{2s}}{g}} &\geqslant & \frac{\mathrm{m}}{\mathrm{QB}_y} \\ \mathrm{Q} &\geqslant & \frac{\mathrm{m}}{\mathrm{B_{y}}}\sqrt{\frac{\mathrm{g}}{\mathrm{2}\mathrm{s}}} \ \ \ \mathrm{Inequality (1)}\end{array}Accordingly, if we wanted complete deflection of the phone before impact with the ground we could calculate the charge that should appear on an outer plate of the falling phone using  inequality (1). 

For a 'Nokia C2 2nd Edition' smartphone, as a random example, the mass of the phone, m = 0.14kg (i.e. 140 grams). It's face area is 14.3⋅10^-2m * 7.14⋅10^-2m. A value we could use for B_y is the upper estimate given in a Wikipedia article; “Earth’s magnetic field”, as 65𝛍T at the Earth's surface. And assuming the phone drops from an altitude of 2m we plug that in for s in inequality (1) as the vertical distance traveled. The charge we’d need on the plate would  therefore be ⩾ 3373.02C. 

To converge upon the values we’d need in the capacitor's specifications we consider that in general the charge, Q, on a plate of a parallel plate capacitor is given by,

\begin{array}{rcl} \mathrm{Q} & = & \frac{\mathrm{A}\varepsilon\mathrm{V}}{\mathrm{d}} \ \ \ \mathrm{Equation (5)} \\ so \ \ \varepsilon &= &\frac{\mathrm{Qd}}{\mathrm{AV}} \ \ \ \mathrm{Equation (6)} \end{array} where A is the area of the capacitor’s plates for which we'll use the face area of the phone given above, 𝛆 is the permittivity of it’s dielectric, d is the separation of the plates which we'll assume to be 0.25⋅10^-3m and V is the voltage across the capacitor. We’ll presume an achievable 12 volts for V. 

Plugging in all our assumed parameters into equation (6), for the Q we calculated above we’d need a dielectric with a permittivity, 𝛆 ⩾ 6.882 F/m or a relative permittivity (dielectric constant), 𝛆_r ⩾ 7.77⋅10¹¹. Now, from googling I discovered that the highest value for 𝛆_r for dielectrics currently synthesised is in the order of 250000 i.e. for calcium copper titanate. So we’re still behind on our dielectrics to make the cushioning I propose possible.

Be that as it may, since research is ongoing in developing dielectrics of higher 𝛆_r, I wouldn’t rule out the subsystem I propose as an option fundis in this arena could one day bring to reality. I’m also open to the possibility that this concept could be employed for purposes other than cushioning electronic gadgets.