Capacitance of parallel plate capacitor with conducting and dielectric slab.

Capacitance of parallel plate capacitor with conducting and dielectric slab.

Capacitance of a parallel plate capacitor with a conducting slab

Let A is the area of plates and d is the separation between them is d then capacitance C0= ϵ0A/d .

Let Q is the charge on the plates of the capacitor. When a conducting slab of thickness ‘t’  ( t< d ) is inserted between the plates . Then the original electric field E0 exist over a distance (d-t) , And inside the metallic plate electric field is zero . so net potential difference between the plates is

V = E0 (d – t) = σ(d-t)/ϵ0 ;   where σ=Q/A

As C = Q/V    = Q /( σ(d-t)/ϵ0 ) ;  But , σ = Q/A

C = Aϵ0 /(d-t) = Aϵ0/[d (1-t/d)] = C0/(1-t/d)

Special case if    t = d  then   new capacitance of the arrangement will be  infinity ,

Capacitance of a parallel plate capacitor with a dielectric slab  –

Let A is the area of plates and d is the separation between them is d then capacitance C0= ϵ0A/d .

Let Q is the charge on the plates of the capacitor. When a dielectric slab of thickness ‘t’  ( t< d ) , and dielectric constant K  is inserted between the plates . Then the original electric field E0 exist over a distance (d-t) , And inside the dielectric plate electric field is E0/K . So net potential difference between the plates is ,

V = E0 (d – t) +( E0 t/K)  = E0[d-t+(t/K)]

= σ[d-t+(t/K)] /ϵ0  ,   where σ=Q/A

As C = Q/V    = Q / σ[d-t+(t/K)] /ϵ0  ;  But , σ = Q/A

C = Aϵ0 /(d-t +t/K)= Aϵ0/[d (1-t/d+t/dK)] = C0/(1-t/d  +t/dK) ;

If t= d   then  new capacitance  C = KC0

 

*** Special cases –

When a dielectric  slab of dielectric constant ( K ) is inserted between the plates having same thickness of separation between the plates . (t = d ) . 

 

When battery disconnected             

( I ) Capacitance C = KC0

(II) Potential  V =  V0/K

(III) Charge  Q = Q0

(iv) potential energy U = U0/K

  When battery remain connected       

(I) Capacitance C = K C0

(II) Potential  V = V0

(III)  Charge   Q = K Q0

(IV)  potential energy  U = KU0

 

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