E The surface charge approaches the equilibrium value E(at which point, from(3) En=0)but it takes a time of the order of t=ok to reach this equilibrium. For a concentrated ionic solution, with K-1 Si/m and e-100, this time is about T=10-5=1 ns, which is difficult to measure directly, but has measurable consequences in the dynamics of very small liquid flows, as we will see. For normal clean"water, K-104 Si/m, and t-10s= 10 us which can be directly measured in the lab The math can be generalized to a gradual variation of the field, E9=En(t). Using the method of variation of the constant o,=c(t)e/i dtdt t and substituting into(4) dc_e/KEg(t); C=Co+e/E(t)dt Since o(0)=0,c(0)=0 And So Co=0 o,=e En(t )dt 16.522, Space Propulsion ure23-25 Prof. Manuel martinez-Sanchez16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 5 of 36 g t - n f 0 E = 1-e τ ⎛ ⎞ σ ⎜ ⎟ ε ⎝ ⎠ (5) The surface charge approaches the equilibrium value g n 0 E ε (at which point, from (3), l E =0 n ) but it takes a time of the order of = 0 K εε τ to reach this equilibrium. For a concentrated ionic solution, with K 1 Si m ∼ and ε ∼ 100 , this time is about -9 τ = 10 s = 1 ns , which is difficult to measure directly, but has measurable consequences in the dynamics of very small liquid flows, as we will see. For normal “clean” water, -4 K 10 Si m ∼ , and -5 τ ∼ 10 s = 10 sµ which can be directly measured in the lab. The math can be generalized to a gradual variation of the field, ( ) g g E Et n n = . Using the method of “variation of the constant” ( ) t - f =c t e τ σ ; t - d f dc c = -e dt dt τ σ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ τ and substituting into (4), ( ) t g n dc K =e E t dt τ ε ; ( ) t t' g 0 n 0 K c = c + e E t' dt' τ ε ∫ Since σf (0 =0 ) , c(0) = 0 And so 0 c =0 : ( ) t t-t' - g f n 0 K = e E t' dt' τ σ ε ∫ (6)