JEE Main Physics Class 12 Electrostatic Potential and Capacitance Part 1 Questions

Consider an infinitesimal strip of capacitor of thickness dx at a distance x as shown.
Capacitance of parallel plate capacitor of area A is given by C=

ε0A

t

[Here t = seperation between plates]
So, capacitance of thickness dx will be
∴dC=

ε0adx

d+x‌tan‌α

Total capacitance of system can be obtained by integrating with limits x = 0 to x = a
∴Ceq=∫dC=aε0‌

x=a

∫

x=0

dx

x‌tan‌α+d

[By Binomial expansion]
⇒Ceq=

aε0

d

‌

a

∫

0

(1−

x‌tan‌α

d

)dx =

aε0

d

(x−

x2‌tan‌α

2d

)0a
⇒Ceq=

a2ε0

d

=(1−

a‌tan‌α

2d

) =

ε0a2

d

(1−

αa

2d

)