Magnetic field at the center of a circular current carrying coil

Magnetic field at the center of a circular current carrying coil is the one of the application of Biot-Savart’s law . here we will derive the expression for Magnetic field at the center of a circular current carrying coil .

syllabus class 12th physics (2020-2021)

Magnetic field at the center of a circular  current carrying coil – Consider a circular coil of radius ‘r’ having center ‘O’. suppose I be the current flowing through the coil , and we have to find the magnetic field at the center .

Suppose a small element ‘dl’ which is the part of coil create a magnetic field dB at the center.

According to Biot-savart’s law dB = (µ0/4Π) I dl sinθ/r2 . but ϴ=900,

So we can write dB = (µ0/4Π) I dl sin900/r2 = dB = (µ0/4Π) I dl /r2 .

Then magnetic field at the center due to complete coil

B=∫ (µ0/4Π) I dl sinθ/r2 ( Taking limit 0 to 2∏)

We get B= dB = (µ0/4Π) I 2∏r/r2  =  µ0 I /2r

For an arc which is making angle ϴ at the center will be given as

B= (µ0 I /4∏r)(Angle at the center )

Or , B= (µ0 Iϴ /4∏r) ;

The direction of magnetic field due to current carrying coil may be give by right hand rule , according to it if curled finger shows the direction of current then stretched thumb gives the direction of magnetic field .