Cosine varies between -1 and 1. cos[√(1/LC-(R/2L)²)t]=1 when t=0 or when:
1/LC-(R/2L)²=0, 1/LC=R²/4L², R²=4L/C; and cos[√(1/LC-(R/2L)²)t]=-1 when:
√(1/LC-(R/2L)²)t=π. Therefore R²=4×50000=200000, R=447.21 ohms. R cannot exceed this value or the square root could not be evaluated. R/2L≤447.21/10=44.721.
When R=0, q=q₀cos(t√2000); when R=447.21, q=q₀e^-44.721t.
When q/q₀=0.01=e^-44.721t, -4.6052=-44.721t, t=0.3256s.
If t=0.05, 0.01=e^(-0.005R)cos[0.05√(2000-0.01R²)].
Consider f(R)=e^(-0.005R)cos[0.05√(2000-0.01R²)]-0.01. We need two values of R (R₁ and R₂) such that f(R₁)<0 and f(R₂)>0. Since R≤447.21, let’s try R₁=300 and R₂=400.
f(300)=-0.0295 and f(400)=0.0631. So we know that 300<R<400.
The table below shows a range of values of R between 300 and 400:
R f(R)
300 -0.0295
320 -0.0082
325 -0.0032
330 0.0018
There is a change of sign between 325 and 330, so 325<R<330.
R f(R)
325 -0.0032
326 -0.0021
327 -0.0011
328 -0.0002
329 0.0008
328<R<329
R f(R)
328.0 -0.00015
328.2 0.00005
R f(R)
328.00 -0.0001507
328.15 -0.0000014
328.16 0.0000085
R f(R)
328.150 -0.000001422
328.151 -0.000000427
328.152 0.000000568
R f(R)
328.1510 -0.000000427
328.1512 -0.000000228
328.1513 -0.000000128
328.1514 -0.000000029
328.1515 0.000000071
R f(R)
328.15140 -0.00000002894
328.15142 -0.00000000904
328.15143 0.00000000091
So R=328.15143 to 5 decimal place accuracy.
Now compare that with the Regula-Falsi Method:
We can work out the equation of the line that joins the two points (300,-0.02950255687) and (400, 0.0631219656):
Slope=(0.0631219656+0.02950255687)/100=0.00092624522.
Line is y-0.0631219656=0.00092624522(x-400). When y=0, 0.00092624522(x-400)=-0.0631219656, x=400-0.0631219656/0.00092624522=331.8517776. So R₁=331.8517776.
We know from the above method that f(R₁)>0.
Therefore 300<R<R₁.
f(R₁)=0.00365859728, f(300)=-0.02950255687.
Slope between the points is (0.00365859728+0.02950255687)/31.8517776=0.0010410843.
Line is y+0.02950255687=0.0010410843(x-300), so x=300+0.02950255687/0.0010410843=328.3382977. R₂=328.3382977.
f(R₂)=0.00018593343, f(300)=-0.02950255687.
Line is
y+0.02950255687=(0.00018593343+0.02950255687)(x-300)/28.3382977=
0.0010476455(x-300).
x=300+0.02950255687/0.0010476455=328.1608205. R₃=328.1608205.
f(R₃)=0.0000093474, f(300)=-0.02950255687.
Line is
y+0.02950255687=(0.0000093474+0.02950255687)(x-300)/28.1608205=
0.00104797743(x-300).
x=300+0.02950255687/0.00104797743=328.15190102.
R₄=328.15190102.
f(R₄)=0.00000046973.
y+0.02950255687=(0.00000046973+0.02950255687)(x-300)/28.15190102=
0.00104799411(x-300).
x=300+0.02950255687/0.00104799411=328.1514528.
R₅=328.1514528.
R₆=328.15143 to 5 decimal places. Compare with 328.15143 obtained earlier.
