JEE Main & Advanced Physics Magnetic Effects of Current / करंट का चुंबकीय प्रभाव Magnetic Field Due to Circular Current

If a coil of radius \[r,\] carrying current \[i\] then magnetic field on it's axis at a distance \[x\] from its centre given by (Application of Biot-Savart's law)

(1) \[{{B}_{axis}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{2\pi Ni{{r}^{2}}}{{{({{x}^{2}}+{{r}^{2}})}^{3/2}}};\] where N = number of turns in coil.

(2) At centre \[x=0\Rightarrow \]\[{{B}_{centre}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{2\pi Ni}{r}\]\[=\frac{{{\mu }_{0}}Ni}{2r}={{B}_{\max }}\]

(3) The ratio of magnetic field at the centre of circular coil and on it's axis is given by \[\frac{{{B}_{centre}}}{{{B}_{axis}}}={{\left( 1+\frac{{{x}^{\mathbf{2}}}}{{{r}^{\mathbf{2}}}} \right)}^{\mathbf{3/2}}}\]

(4) If \[x>>r\Rightarrow \] \[{{B}_{axis}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{2\pi \,Ni{{r}^{2}}}{{{x}^{3}}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{2NiA}{{{x}^{3}}}\]

where \[A=\pi {{r}^{2}}=\] Area of each turn of the coil.

(5) B-x curve : The variation of magnetic field due to a circular coil as the distance x varies as shown in the figure.

B varies non-linearly with distance x as shown in figure and is maximum when \[{{x}^{2}}=\min =0\], i.e., the point is at the centre of the coil and it is zero at \[x=\pm \,\infty \]. .

(6) Point of inflection (A and A') : Also known as points of curvature change or points of zero curvature.

(i) At these points B varies linearly with \[x\Rightarrow \frac{dB}{dx}=\] constant \[\Rightarrow \frac{{{d}^{2}}B}{d{{x}^{2}}}=0\].

(ii) These are located at \[x=\pm \frac{r}{2}\] from the centre of the coil and the magnetic field at \[x=\frac{r}{2}\] is \[B=\frac{4{{\mu }_{0}}Ni}{5\sqrt{5}\,r}\]

(7) Helmholtz coils

(i) This is the set-up of two coaxial coils of same radius such that distance between their centres is equal to their radius.

(ii) At axial mid point O, magnetic field is given by \[B=\frac{8{{\mu }_{0}}Ni}{5\sqrt{5}R}=0.716\frac{{{\mu }_{0}}Ni}{R}=1.432\,B\], where \[B=\frac{{{\mu }_{0}}Ni}{2R}\]

(iii) Current direction is same in both coils otherwise this arrangement is not called Helmholtz's coil arrangement.

(iv) Number of points of inflextion \[\Rightarrow \] Three \[(A,\,A',\,A'')\]