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Sensors 2020, 20, 6087 11 of 13 Solving Ex for when y = 1 μm (a minor offset is required within the unit-vector conversion). The units used will be millimeters. y2 is negligible when compared to x and z, as x will be on the scale of millimeters. Ex=− x Simplified, tan −1 z2 √ κ x2+z2 2− √2 κ x +z 2e √ −jκ x+z (A14) (A15) 0.001CI(ω,t) x2 j 1 √2 2 Ex =−CI(ω,t) √ κx+z 1000xzx2+1 z j κx2+z2 1 |x|e−jκ x2+z2 2 − √ 2 2 2 Similarly converting the time derivative of the vector potential to Cartesian coordinates and extracting the x-component, | x | μ I ( ω , t ) R 2 | ( x 2 + y 2 | x 2 + y 2 + z 2 0 √ 2 2 2 −jκ x+y+z ∂ A ̄ ∂t x 4|z| z2 +1(R +(x +z )) Finally, the activating potential is obtained from =− jω e (A16) (A17) (A18) x2 |y| y +1 4|z| x2+y2 +1(R2 +(x2 +y2 +z2))3/2 z2 μ I ( ω , t ) R 2 | x | √ x 2 + z 2 0 √ 2 −jκ x+z ∂ A ̄ ∂tx=−jω x2 2 2 2 3/2e 2 ̄ ̄δEx+∂A AF(x)=−∇ Ex(x)+∂A(x) =−xˆ ∂tx ∂t x δx The activation function results demonstrate the capability of the microcoils to stimulate excitable tissue in the cochlea. Any neurons with axons arranged parallel with the generated electric fields will demonstrate stimulation represented by the activating function. Depolarization occurs when AF(x) > 0, and hyperpolarization occurs when AF(x) < 0. The activating function provides a means to observe spatial resolution along one dimension and can be further manipulated to observe the general spatial resolution of the microcoils. Table A1. Units for electromagnetic properties. Symbol B H D E V κ J ρ μ μr ε εr Quantity magnetic flux density magnetic field strength electric displacement electric field strength scalar electrostatic potential wavenumber current density charge density permeability relative permeability permittivity relative permittivity Units T A/m C·m2 V/m V Radians/m A/m3 C/m3 4π × 10−7 · μr H/m Unitless 8.85 × 10−12 · εr F/m UnitlessPDF Image | Silver-Nanoparticle Microcoil via Aerosol Jet Printing
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