Answer:

Magnetic INDUCTION due to infinitely long straight conductor carrying current: XY, is an infinitely long straight conductor carrying a current I(Figure).

P is a point at a distance a from the conductor. AB is a small element of length dl. θ is the angle between the current element Idl and the line joining the element dl and the point P. According to Biot-Savart law, the magnetic induction at the point P due to the current element Idl is.

dB=

4π

μ

o

r

2

Idl⋅sinθ

......(1)

AC is drawn perpendicular to BP from A.

∠OPA=ϕ, ∠APB=dϕ

In ΔABC, sinθ=

AB

AC

=

dl

AC

∴AC=dlsinθ ..........(2)

From ΔAPC, AC=rdϕ .........(3)

From equations (2) and (3),

rdϕ=dlsinθ ............(4)

Substituting equation (4) in equation (1)

dB=

4π

μ

o

r

2

Irdϕ

=

4π

μ

o

r

Idϕ

.............(5)

In ΔOPA, cosϕ=

r

a

∴r=

cosϕ

a

..........(6)

Substituting equation (6) in equation (5)

dB=

4π

μ

o

a

1

cosϕdϕ

The total magnetic induction at P due to the conductor XY is

B=

−ϕ

1

∫

ϕ

2

dB=

−ϕ

1

∫

ϕ

2

4πa

μ

o

I

cosϕdϕ

B=

4πa

μ

o

I

[sinϕ

1

+sinϕ

2

]

For infinitely long conductor,

ϕ

1

=ϕ

2

=90

o

∴B=

2πa

μ

o

I

If the conductor is places in a medium