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Question
Use Biot-Savart law to derive the expression for the magnetic field on the axis of a current carrying circular loop of radius R.
Draw the magnetic field lines due to a circular wire carrying current I.
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Solution
I = Current in the loop
R = Radius of the loop
X-axis = Axis of the loop
X = Distance between O and P
dl = Conducting element of the loop
According to the Biot–Savart law, the magnetic field at P is
`dB=(mu_0)/(4pi) (I|dlxxr|)/r^3`
r2 = x2 + R2
|dl × r| = rdl (Because they are perpendicular)
`:.dB=mu_0/(4pi) (Idl)/((x^2+R^2))`
dB has two components: dBx and dB⊥. dB⊥ is cancelled out and only the x-component remains.
∴ dBx= dBcos θ
`costheta= R/(x^2+R^2)^(1/2)`
`:.dB_x=(mu_0Idl)/(4pi) R/(x^2+R^2)^(3/2)`
Summation of dl over the loop is given by 2πR.
`:.B=B=B_xhati=(mu_0IR^2)/(2(x^2+R^2)^(3/2))hati`
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