magnetic field due to current through a very long circular cylinder
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If the current density as a function of distance 'r' from the axis of a radially symmetrical parallel stream of electrons is given as j(r)=
μ
0
xb(α+1)
r
α−1
if the magnetic induction inside the stream varies as B=br
α
, where b and α are positive constants. Find 'x'
Using Ampere's Circuital Law,∮
p
B.dl=μ
o
I (Equation 1)
Total current (I) flowing through the cross-section of radius r, I=∫
0
r
j(r
′
)×A
I=∫
0
r
μ
o
xb(α+1)(r
′
)
α−1
2πr
′
dr
′
I=
μ
o
xb(α+1)
2π∫
0
r
(r
′
)
α
dr
′
I=
μ
o
xb(α+1)
2π
α+1
r
α+1
I=
μ
o
bx2πr
α+1
∮
p
B.dl=∮
p
br
α
dl=br
α
∮
p
dl
∮
p
B.dl=br
α
2πr=2πbr
α+1
From equation 1 , 2πbr
α+1
=μ
o
μ
o
bx2πr
α+1
Hence, x=1
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