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 circular wire carrying current I.
Updated On 18th June 2025
Expert Explanation :
A current carrying wire produces a magnetic field around it. Biot-Savart law states that magnitude of intensity of small magnetic field $d \vec{B}$ due to current I carrying element $d \vec{l}$ at any point $P$ at distance $r$ from it is given by
$
|d \vec{B}|=\frac{\mu_0}{4 \pi} \frac{\mid d \vec{l} \sin \theta}{r^2}
$
where $\theta$ is the angle between $\vec{r}$ and $d \vec{l}$ and $\mu_0=4 \pi \times 10^{-7} \mathrm{~T} \mathrm{~m} \mathrm{~A}^{-1}$ is called permittivity of free space. In vectorial form,
$
d \vec{B}=\frac{\mu_0}{4 \pi} \frac{d \vec{l} \times \vec{r}}{r^3}
$
So, the direction of $d \vec{B}$ is perpendicular to the plane containing $\vec{r}$ and $d \vec{l}$.
.png)
S.I. unit of magnetic field strength is tesla denoted by ' $T$ ' and cgs unit is gauss denoted by ' G ', where $1 \mathrm{~T}=10^4 \mathrm{G}$ Magnetic field on the axis of a circular coil
Small magnetic field due to a current element of circular coil of radius $r$ at point $P$ at distance $x$ from its centre is
$
d B=\frac{\mu_0}{4 \pi} \frac{I d \vec{l} \sin 90^{\circ}}{s^2}=\frac{\mu_0}{4 \pi} \frac{\mid d \vec{l}}{\left(a^2+x^2\right)}
$
.png)
Component $d B \cos \phi$ due to current element at point $P$ is cancelled by equal and opposite component $d B \cos \phi$ of another diametrically opposite current element, whereas the sine components $d B \sin \phi$ add up to give net magnetic field along the axis. So, net magnetic field at point $P$ due to entire loop is $B=\oint d B \sin \phi=\int_0^{2 \pi r} \frac{\mu_0}{4 \pi} \frac{l d \vec{l}}{\left(a^2+x^2\right)} \cdot \frac{r}{\left(a^2+x^2\right)^{1 / 2}}$
$
B=\frac{\mu_0 l r}{4 \pi\left(a^2+x^2\right)^{3 / 2}} \int_0^{2 \pi r} d l
$
or $B=\frac{\mu_0 l r}{4 \pi\left(a^2+x^2\right)^{3 / 2}} 2 \pi r$
or $B=\frac{\mu_0 l r^2}{2\left(a^2+x^2\right)^{3 / 2}}$ directed along the axis,
(a) towards the coil if current in it is in clockwise direction
(b) away from the coil if current in it is in anticlockwise direction.
Magnetic field lines due to circular wire carrying current I:
.png)
Key Points
- Magnetic field lines around a current carrying wire are up when the wire is bent into circular loop.
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