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Question
Use Biot-Savart's law to find the expression for the magnetic field due to a circular loop of radius 'r' carrying current 'I', at its centre ?
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Solution
Let us consider a circular coil of radius r with centre at O, carrying current I in the direction as shown in the figure. Consider that the coil is made up of large number of current elements each of length dl.
According to Biot-Savart's law, the magnetic field at the centre O of the coil will be
\[{dB}^\rightharpoonup = \frac{\mu_0}{4\pi}I\left( \frac{{dI}^\rightharpoonup \times r^\rightharpoonup}{r^3} \right)\]
\[ \Rightarrow {dB}^\rightharpoonup = \frac{\mu_0}{4\pi}\left( \frac{Idlr\sin\theta}{r^3} \right)\]
\[ \Rightarrow {dB}^\rightharpoonup = \frac{\mu_0}{4\pi}\left( \frac{Idl\sin\theta}{r^2} \right)\]
\[, \text { where} \ \vec{r} \ \text {is the position vector of O from current element dl } . \]
\[\text { Since the angle between dl and r }, \theta = {90}^0 \]
\[ \therefore \vec{dB} = \frac{\mu_0}{4\pi}\left( \frac{Idlsin {90}^0}{r^2} \right)\]
\[ \Rightarrow {dB}^\rightharpoonup = \frac{\mu_0}{4\pi}\left( \frac{Idl}{r^2} \right)\]
\[\text { The direction of } {dB}^\rightharpoonup \text { is perpendicular to the plane and directed inwards } . \]
\[\text{ Since the magnetic field due to each element is in same direction, the net magnetic field can be integrated as } \]
\[B = \int dB = \int\frac{\mu_0}{4\pi}\frac{Idl}{r^2} = \frac{\mu_0 I}{4\pi r^2}\int dl\]
\[\int dl = \text { Total length of circular coil }= 2\pi r (\text { Circumference of coil }) \]
\[ \Rightarrow B = \frac{\mu_0 I}{4\pi r^2}2\pi r = \frac{\mu_0 I2\pi}{4\pi r}\]
