870 The plates of a capacitor are not quite parallel, the distance between them at the bottom being (D a) and at the top (Dta), where a
Solution
let length of each plate be L and width be W.
then area A=L*W
at a distance x from the bottom end of the plate, distance between the plates as a function of x is given as
d=D-a+(2*a/L)*x
consider a small segment of length dx at a distance of x from the bottom end
area of the segment=W*dx
distance between the two plates=d=(D-a+(2*a/L)*x)
then capacitance=dC=epsilon*area/distance between plates
=epsilon*W*dx/(D-a+(2*a/L)*x)
let D-a+(2*a/L)*x=y
==>(2*a/L)*dx=dy
==>dx=L*dy/(2*a)
substituting,
dC=epsilon*W*L*dy/(2*a*y)
integrating both sides,
C=(epsilon*W*L/(2*a))*ln(y)
as y varies from D-a to D+a
C=(epsilon*W*L/(2*a))*ln((D+a)/(D-a))
as A=W*L
and ln((D+a)/(D-a))
=ln((1+(a/D))/(1-(a/D)))
=2*(a/D)*(1+(a^2/(3*D^2)))
then C=(epslion*A/(2*a))*(2*a/D)*(1+(a^2/(3*D^2)))
=(epsilon*A/D)*(1+(a^2/(3*D^2)))
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870 The plates of a capacitor are not quite parallel- the distance bet.docx
1. 870 The plates of a capacitor are not quite parallel, the distance between them at the bottom
being (D a) and at the top (Dta), where a
Solution
let length of each plate be L and width be W.
then area A=L*W
at a distance x from the bottom end of the plate, distance between the plates as a function of x is
given as
d=D-a+(2*a/L)*x
consider a small segment of length dx at a distance of x from the bottom end
area of the segment=W*dx
distance between the two plates=d=(D-a+(2*a/L)*x)
then capacitance=dC=epsilon*area/distance between plates
=epsilon*W*dx/(D-a+(2*a/L)*x)
let D-a+(2*a/L)*x=y
==>(2*a/L)*dx=dy
==>dx=L*dy/(2*a)
substituting,
dC=epsilon*W*L*dy/(2*a*y)
integrating both sides,
2. C=(epsilon*W*L/(2*a))*ln(y)
as y varies from D-a to D+a
C=(epsilon*W*L/(2*a))*ln((D+a)/(D-a))
as A=W*L
and ln((D+a)/(D-a))
=ln((1+(a/D))/(1-(a/D)))
=2*(a/D)*(1+(a^2/(3*D^2)))
then C=(epslion*A/(2*a))*(2*a/D)*(1+(a^2/(3*D^2)))
=(epsilon*A/D)*(1+(a^2/(3*D^2)))
