2.17 (a) Differentiating Equation 2.12 with respect to...
Transcript of 2.17 (a) Differentiating Equation 2.12 with respect to...
2-19
2.17 (a) Differentiating Equation 2.12 with respect to r yields
dEdr
= d −
Cr
⎛ ⎝ ⎜
⎞ ⎠ ⎟
dr −
d D exp −rρ
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
dr
= C
r2 − De−r /ρ
ρ
At r = r0, dE/dr = 0, and
Cr0
2 = De−(r0/ρ)
ρ (2.12b)
Solving for C and substitution into Equation 2.12 yields an expression for E0 as
E0 = De− (r0/ρ) 1 −
r0ρ
⎛
⎝ ⎜
⎞
⎠ ⎟
(b) Now solving for D from Equation 2.12b above yields
D = Cρ e (r0/ρ)
r02
Substitution of this expression for D into Equation 2.12 yields an expression for E0 as
E0 = C
r0
ρr0
− 1⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
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