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Please answer the following question with explanation

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Asked by Deepak 4th October 2018, 7:44 PM
Answered by Expert
Answer:
Magnetic field B inside capacitor due to displacement current Id is given by, begin mathsize 12px style contour integral B times d l space equals space mu subscript 0 I subscript d space equals space mu subscript 0 epsilon subscript 0 fraction numerator d phi over denominator d t end fraction end style.....................(1)
where μ0 is permeability of free space, ε0 is permitivity of free space, φ is flux of electric field.
 
from eqn.(1), we can write that, begin mathsize 12px style B space equals space fraction numerator mu subscript 0 epsilon subscript 0 over denominator 2 pi r end fraction fraction numerator d phi over denominator d t end fraction space.............. left parenthesis 2 right parenthesis end style
where r is the radius of capacitor plate.
Flux φ = Electric field × area of capacitor plate = begin mathsize 12px style fraction numerator V o l t a g e space a p p l i e d space t o space t h e space p l a t e s over denominator d i s tan c e space b e t w e e n space p l a t e s end fraction cross times A r e a space o f space c a p a c i t o r space p l a t e end style ...........................(3)
flux φ(t)  = begin mathsize 12px style fraction numerator 2 square root of 2 sin left parenthesis 2 pi cross times 50 cross times 1000 cross times t right parenthesis over denominator d end fraction cross times A end style .......................................(4)
let us differentiate eqn.(4) and substitute in eqn.(2) to get B
begin mathsize 12px style B space equals space fraction numerator mu subscript 0 over denominator 2 pi space r end fraction cross times fraction numerator epsilon subscript 0 A over denominator d end fraction cross times space 2 cross times square root of 2 cross times 10 to the power of 5 cross times pi space cross times cos left parenthesis 2 pi cross times 50 cross times 1000 cross times t right parenthesis space end style ...........................................(5)
In eqn.(5), begin mathsize 12px style fraction numerator epsilon subscript 0 A over denominator d end fraction end style is the capacitance.  By substituting the capacitance value 10μF,   r = 10 cm, μ0 = 4π×10-7  and
considering maximum value of cosine function is one, we get the maximum value of B = 4√2 π × 10-6 T
Answered by Expert 5th October 2018, 4:38 AM
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