The value of magenetic field produced by coduction current and displacement current are equal at the edges ofor capacitor or not?

Asked by 0987gauranshi | 4th Jan, 2022, 02:33: PM

Expert Answer:

Let us consider charging a parallel plate capacitor as shown in figure.
 
Let us consider an Amperian loop across the conductor .
 
This amperian loop can be boundary to a flat surface or a baloon like surface as shown in figure.
 
In the flat surface , Ampere's law is written as
 
begin mathsize 14px style contour integral B times d l space space equals space mu subscript o I subscript c end style ...................................(1)
where Ic is conduction current.
 
If the balloon like surface encloses only displacement current Id inside capacitor plates then
 
begin mathsize 14px style contour integral B times d l space space equals space mu subscript o I subscript d space equals space mu subscript o epsilon subscript o integral subscript S open parentheses fraction numerator partial differential E over denominator partial differential t end fraction close parentheses d a end style  ........................(2)
Subscript S in the integral represents surface integration performed over baloon like surface

Electric field inside capacitor E is given as
 
begin mathsize 14px style E space equals space sigma over epsilon subscript o space equals space fraction numerator Q over denominator A space epsilon subscript o end fraction end style

where σ is charge density , Q is charge at an instant of time t and A is area of plates
 
begin mathsize 14px style fraction numerator partial differential E over denominator partial differential t end fraction space equals space fraction numerator 1 over denominator A space epsilon subscript o end fraction space fraction numerator partial differential Q over denominator partial differential t end fraction equals space fraction numerator 1 over denominator A space epsilon subscript o end fraction space I subscript c end style.......................................(3)
 
From eqn.(2) and eqn.(3) , we get
 
begin mathsize 14px style contour integral B times d l space space equals space mu subscript o I subscript c end style ...............................(4)
Above equation is same as eqn.(1)
 
Hence magnetic field produced by conduction current and magnetic field produced  by
displacement current at the edge of capacitor are same
 



 

Answered by Thiyagarajan K | 4th Jan, 2022, 05:10: PM