Calculate the capacitance of the following multi slab capacitor. 

Asked by hitanshu04 | 21st May, 2021, 07:05: PM

Expert Answer:

Let σ be the surface charge density on capacitor plates .
If a dielctric slab of dielectric constant K and thickness d is inside capacitor plates ,
Then electric field E in the dielectric slab is  given as
 
begin mathsize 14px style E space equals space fraction numerator sigma over denominator K space epsilon subscript o end fraction end style  .............................. (1)
 
Potential difference across dielectric slab, V  = E × d =  begin mathsize 14px style fraction numerator sigma over denominator K space epsilon subscript o end fraction cross times d end style ........................(2)
 
Potential difference across capacitor plates having three different dielectric slabs is given as
 
begin mathsize 14px style V space equals space V subscript 1 space plus space V subscript 2 space plus space V subscript 3 space end subscript space equals space fraction numerator sigma over denominator K subscript 1 space epsilon subscript o end fraction cross times d subscript 1 space plus space space fraction numerator sigma over denominator K subscript 2 space epsilon subscript o end fraction cross times d subscript 2 space plus space space fraction numerator sigma over denominator K subscript 3 space epsilon subscript o end fraction cross times d subscript 3 end style ..........................(3)

If Q is the total charge on capacitor plates and A is area of plates , then surface charge density σ = (  Q / A  )
 
By substituting surface charge density in eqn.(3) , we get
 
begin mathsize 14px style V space equals space Q over A space open parentheses fraction numerator d subscript 1 over denominator K subscript 1 space epsilon subscript o end fraction space plus space space fraction numerator d subscript 2 over denominator K subscript 2 space epsilon subscript o end fraction space plus space space fraction numerator d subscript 3 over denominator K subscript 3 space epsilon subscript o end fraction close parentheses end style............................. (4)
Using the definition of capacitance C = ( Q / V ) , we get capacitance of multislab dielectric capacitor as

begin mathsize 14px style C thin space equals space Q over V space equals space fraction numerator A space epsilon subscript o space over denominator open parentheses fraction numerator d subscript 1 over denominator K subscript 1 space end fraction space plus space space fraction numerator d subscript 2 over denominator K subscript 2 space end fraction space plus space space fraction numerator d subscript 3 over denominator K subscript 3 space end fraction close parentheses end fraction space end style

Answered by Thiyagarajan K | 21st May, 2021, 08:55: PM