Off[General::spell1]; Off[General::spell]

{Pmm, ΔPmm, p, p, Δp,  
          d,  Δd, Voltage, ΔVoltage, 
          R, ΔR, Ao, Bo, Co, To, Tc} =. ;

This worksheet calculates the charge on a droplet from the UP and DOWN terminal speeds with the field applied for both motions. (Note: Required input values are in red) To evaluate all cells after entering all values go to "Kernel :  Evaluation : Evaluate Notebook".

Error propagation

This function is used multiple times in the worksheet to find uncertainty so we define it here.

delta[x_, Δx_ , 
        y_, Δy_ ] := (Sqrt[(((Δ(x/x)))^2) + (((Δ(y/y)))^2)])

Apparatus Values

Room pressure

p =  Pmm (((1.013×(10^5))/760) (Pa/mmHg))

Δp = p  Δ(Pmm/Pmm)

Oil density and other parameters

Capacitor

plate seperation (d) and uncertainty (Δd)

plate voltage and uncertainty

v := ((kg (m^2))/(C (s^2)))

Fit for Temperature

resistance of thermistor (R in MΩ) and uncertainty (ΔR in MΩ)

resistance := {1.496, 1.521, 1.547, 1.574, 1.603, 1.634, 1.666, 1.700, 
        1.736, 1.774, 1.815, 1.857, 1.902, 1.950, 2.000, 2.053, 2.110, 2.169, 
        2.233, 2.300, 2.371, 2.446, 2.526, 2.610, 2.700, 2.795, 2.897, 3.004, 
        3.118, 3.239}

temperature := {39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 
        25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 
        10}

interp := 
      Interpolation[Transpose[{resistance, temperature}]]

chamber temperature Tc[R]

Tc = interp[R] C

uncertainty in Tc[R]

ΔTc = 
      Abs[interp'[R ] ΔR] C

Viscosity of air (η)

η = ((4.765×(10^(-3)) (C^(-1))) Tc + 
            1.729) (10^(-5)) kg (m^(-1)) (s^(-1))

Δη := 
      4.765×(10^(-8)) (kg/(C m s)) ΔTc

Electric field (e)

e := (Voltage/d); Δe := 
      e delta[Voltage, ΔVoltage, 
          d, Δd];

Droplet Speeds

Terminal up speed (Vu)

Vu = (Su/Tu);  ΔVu = 
      Vu delta[Su, ΔSu, 
          Tu, ΔTu];

Terminal down speed (Vd)

Vd = (Sd/Td);  ΔVd = 
      Vd delta[Sd, ΔSd, 
          Td, ΔTd];

Results

Droplet radius (a)

A[press_, visc_, vdn_, 
        vup_] := (Sqrt[((((b/(2 press))))^2) + ((9 visc (vdn - vup))/(4 g ρ))]) - (b/(2 press))

a = Simplify[A[p, η, Vd, Vu], m ≥ 0]

Derivatives of function A with respect to each variable give the uncertainty

dAp := 
      Simplify[Derivative[1, 0, 0, 0][A][p, η, Vd, Vu], 
        m ≥ 0]

dAη := 
      Simplify[Derivative[0, 1, 0, 0][A][p, η, Vd, Vu], 
        m ≥ 0]

dAVd := 
      Simplify[Derivative[0, 0, 1, 0][A][p, η, Vd, Vu], 
        m ≥ 0]

dAVu := 
      Simplify[Derivative[0, 0, 0, 1][A][p, η, Vd, Vu], 
        m ≥ 0]

Δa = 
      Simplify[(Sqrt[(((dAp Δp))^2) + (((dAη Δη))^2) + (((dAVd ΔVd))^2) + (((dAVu ΔVu))^2)]), m ≥ 0]

Droplet mass

Mass[press_, visc_, vdn_, 
        vup_] := (4/3) π A[press, visc, vdn, 
              vup]^3 ρ

mass = 
      Simplify[Mass[p, η, Vd, Vu], m ≥ 0]

Δmass = 3 Δ(a/a) mass

Charge (q)

Q[press_, visc_, vdn_, vup_, 
        e_] := ((g Mass[press, visc, vdn, 
                vup])/e) ((Vd + Vu)/(Vd - Vu))

q = Simplify[Q[p, η, Vd, Vu, e], m ≥ 0]

Derivatives of function Q with respect to each variable give the uncertainty

dQp := 
      Simplify[Derivative[1, 0, 0, 0, 0][Q][p, η, Vd, Vu, e], 
        m ≥ 0]

dQη := 
      Simplify[Derivative[0, 1, 0, 0, 0][Q][p, η, Vd, Vu, e], 
        m ≥ 0]

dQVd := 
      Simplify[Derivative[0, 0, 1, 0, 0][Q][p, η, Vd, Vu, e], 
        m ≥ 0]

dQVu := 
      Simplify[Derivative[0, 0, 0, 1, 0][Q][p, η, Vd, Vu, e], 
        m ≥ 0]

dQe := 
      Simplify[Derivative[0, 0, 0, 0, 1][Q][p, η, Vd, Vu, e], 
        m ≥ 0]

Δq = 
      Simplify[(Sqrt[(((dQp Δp))^2) + (((dQη Δη))^2) + (((dQVd ΔVd))^2) + (((dQVu ΔVu))^2) + (((dQe Δe))^2)]), {m ≥ 0, C ≥ 0}]