Question

Using Ampere’s Circuital law, drive the expression for magnetic field due to

thick infinite cylindrical wire carrying current.​

Answer 1

Explanation:

According to Ampere's circuital law, integral of magnetic field along a closed curve equals the sum of all electric currents passing through the cross section of the closed curve times permeability.

∮

I

.

dl

=μ

o

I

enclosed

To find magnetic field inside a long straight wire of cross-sectional area a, refer the attached figure. Magnetic field lines in this configuration forms concentric circles centered at O, the axis of the cylindrical wire. Hence, it is the preferred path of the Amperian Loop.

Consider an Amperian Loop of radius a<r as shown in the figure.

Current enclosed in a cross-section area πa

2

is I

Then, Current enclosed in a cross-section area πr

2

is I

enclosed

=

a

2

r

2

I

Taking the magnetic field integral along the loop, we get

∮

B

⋅

dl

=B×2πr

Using Ampere's Law, we get

B×2πr=

a

2

μ

o

Ir

2

B=

2πa

2

μ

o

rI

Direction of magnetic field can be found using Fleming's Right hand thumb rule.