Question
Derive an expression for axial magnetic field produced by current in a circular loop.
✓
Answer
i. Consider loop of radius R carrying current I placed in $x$ - $y$ plane with its centre at origin $O$ as shown in the figure below.
The magnetic field on the axis of a circular current loop of radius $R$
ii. Let point $P$ can be on $z$-axis at distance $\overrightarrow{ r }$ from line element $\overrightarrow{ d } l$ of the loop.
iii. Using Biot-Savart law, the magnitude of the magnetic field $dB$ is given by $dB =\frac{\mu_0}{4 \pi} I \frac{| d \vec{l} \times \overrightarrow{ r }|}{ r ^3}$
iv. Any element $\overrightarrow{ d } l$ will always be perpendicular to the vector $\overrightarrow{ r }$ from the element to the point $P$. The element $\overrightarrow{ d } l$ is in the $x - y$ plane, while the vector $\overrightarrow{ r }$ is in the $y - z$ plane. Hence
$ \overrightarrow{ d } l \times \overrightarrow{ r }= dl r$
$\therefore dB =\frac{\mu_0}{4 \pi} I \frac{ dl }{ r ^2}$
$\text { but } r ^2= R ^2+ z ^2$
$\therefore dB =\frac{\mu_0}{4 \pi} I \frac{ dl }{\left( z ^2+ R ^2\right)} $
v. Now, the direction of $d \overrightarrow{ B }$ is perpendicular to the plane formed by $d \overrightarrow{ l }$ and $\overrightarrow{ r }$. Its $z$ component is $dB _{ Z }$ and the component perpendicular to the $z$-axis is $d_{B_{\perp}}$. The components $dB _{\perp}$ when summed over, yield zero as they cancel out due to symmetry. Hence, only the $z$ component remains.
vi. The net contribution along the $z$-axis is obtained by integrating $dB _{ z }= dB \cos \theta$ over the entire loop. From figure,
$ \cos \theta=\frac{ R }{ r }=\frac{ R }{\sqrt{ z ^2+ R ^2}}$
$\therefore B _{ Z }=\int dB _{ z }=\frac{\mu_0}{4 \pi} I \int \frac{ dl }{\left( z ^2+ R ^2\right)} \cos \theta$
$=\frac{\mu_0}{4 \pi} I \int \frac{ Rdl }{\left( z ^2+ R ^2\right)^{\frac{3}{2}}}$
$=\frac{\mu_0}{4 \pi} \times \frac{ lR }{\left( z ^2+ R ^2\right)^{\frac{3}{2}}} \times 2 \pi R$
$B _{ Z }=\frac{\mu_0}{2} \times \frac{ IR ^2}{\left( z ^2+ R ^2\right)^{\frac{3}{2}}} $
This is the magnitude of the magnetic field due to current $I$ in the loop of radius $R$, on a point at $P$ on the $z$-axis of the loop.
The magnetic field on the axis of a circular current loop of radius $R$
ii. Let point $P$ can be on $z$-axis at distance $\overrightarrow{ r }$ from line element $\overrightarrow{ d } l$ of the loop.
iii. Using Biot-Savart law, the magnitude of the magnetic field $dB$ is given by $dB =\frac{\mu_0}{4 \pi} I \frac{| d \vec{l} \times \overrightarrow{ r }|}{ r ^3}$
iv. Any element $\overrightarrow{ d } l$ will always be perpendicular to the vector $\overrightarrow{ r }$ from the element to the point $P$. The element $\overrightarrow{ d } l$ is in the $x - y$ plane, while the vector $\overrightarrow{ r }$ is in the $y - z$ plane. Hence
$ \overrightarrow{ d } l \times \overrightarrow{ r }= dl r$
$\therefore dB =\frac{\mu_0}{4 \pi} I \frac{ dl }{ r ^2}$
$\text { but } r ^2= R ^2+ z ^2$
$\therefore dB =\frac{\mu_0}{4 \pi} I \frac{ dl }{\left( z ^2+ R ^2\right)} $
v. Now, the direction of $d \overrightarrow{ B }$ is perpendicular to the plane formed by $d \overrightarrow{ l }$ and $\overrightarrow{ r }$. Its $z$ component is $dB _{ Z }$ and the component perpendicular to the $z$-axis is $d_{B_{\perp}}$. The components $dB _{\perp}$ when summed over, yield zero as they cancel out due to symmetry. Hence, only the $z$ component remains.
vi. The net contribution along the $z$-axis is obtained by integrating $dB _{ z }= dB \cos \theta$ over the entire loop. From figure,
$ \cos \theta=\frac{ R }{ r }=\frac{ R }{\sqrt{ z ^2+ R ^2}}$
$\therefore B _{ Z }=\int dB _{ z }=\frac{\mu_0}{4 \pi} I \int \frac{ dl }{\left( z ^2+ R ^2\right)} \cos \theta$
$=\frac{\mu_0}{4 \pi} I \int \frac{ Rdl }{\left( z ^2+ R ^2\right)^{\frac{3}{2}}}$
$=\frac{\mu_0}{4 \pi} \times \frac{ lR }{\left( z ^2+ R ^2\right)^{\frac{3}{2}}} \times 2 \pi R$
$B _{ Z }=\frac{\mu_0}{2} \times \frac{ IR ^2}{\left( z ^2+ R ^2\right)^{\frac{3}{2}}} $
This is the magnitude of the magnetic field due to current $I$ in the loop of radius $R$, on a point at $P$ on the $z$-axis of the loop.
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