- #1

- 590

- 0

_{1}to ε

_{2}, what is the capacitance of C1?

since i know that ε changes linerly, according to the distance between the plates, and i know that at x=0 ε=ε

_{1}and x=d ε=ε

_{2}

ε

_{r}=ax+b

ε

_{r}=ε

_{1}=a*0+b

b=ε

_{1}

ε

_{r}=ε

_{2}=a*d+ε

_{1}

a=(ε

_{2}-ε

_{1})/d

ε

_{r}=(ε

_{2}-ε

_{1})x/d+ε

_{1}

ε

_{r}=((ε

_{2}-ε

_{1})x+ε

_{1}d)/d

know this capacitator C1 is like millions of tiny little capacitators, dC, each with dialectric substance changing like ε

_{r}=((ε

_{2}-ε

_{1})x+ε

_{1}d)/d, all connected in a column,

1/C

_{p}=[tex]\int[/tex]d/(Aε

_{0}ε

_{r})d(ε

_{r}) from (ε

_{1}to ε

_{2})

=d/(Aε

_{0})[tex]\int[/tex]d(d/((ε

_{2}-ε

_{1})x+ε

_{1}d))

=d

^{2}/(Aε

_{0})[tex]\int[/tex]dx/((ε

_{2}-ε

_{1})x+ε

_{1}d) (from 0 to d)

=d

^{2}/(Aε

_{0})*1/(ε

_{2}-ε

_{1})* ln((ε

_{2}-ε

_{1})x+ε

_{1}d)|from 0 to d

=[d

^{2}ln(ε

_{2}/ε

_{1})]/ε

_{0}(ε

_{2}-ε

_{1})Aε

_{0}

but the correct answer is meant to be 1/C

_{p}=[dln(ε

_{2}/ε

_{1})]/ε

_{0}(ε

_{2}-ε

_{1})Aε

_{0}

can anyone see whre i went wrong?