Mathematical Analysis of magnetic field at the center of circular loop:
Let us consider, a current-carrying circular loop of radius $a$ in which $i$ current is flowing. Now take a small length of a current element $dl$ so magnetic field at the center of a circular loop due to the length of current element $dl$. According to Biot-Savart Law:
$dB=\frac{\mu_{\circ}}{4 \pi} \frac{i .dl .sin\: \theta}{a^{2}} $
Here $\theta$ is the angle between length of current element $\left( \overrightarrow{dl} \right)$ and radius $\left( \overrightarrow{a} \right)$. These are perpendicular to each other i.e. $\theta = 90^{\circ}$
$dB=\frac{\mu_{\circ}}{4 \pi} \frac{i .dl .sin\: 90^{\circ}}{a^{2}} $
$dB=\frac{\mu_{\circ}}{4 \pi} \frac{i .dl }{a^{2}} \qquad \left(1 \right)$
The magnetic field at the center due to a complete circular loop
$B=\int dB \qquad \left(2 \right)$
From equation $(1)$ and equation $(2)$
$B=\int \frac{\mu_{\circ}}{4 \pi} \frac{i .dl}{a^{2}}$
$B=\frac{\mu_{\circ}}{4 \pi} \frac{i}{a^{2}} \int dl $
For complete loop $\int dl = 2 \pi a$, So from above equation that can be written as
$B=\frac{\mu_{\circ}}{4 \pi} \frac{i}{a^{2}} \left( 2 \pi a \right) $
$B=\frac{\mu_{\circ} i}{2 a}$
This is an equation of the magnetic field at the center of the circular loop.
Angle of Acceptance → If incident angle of light on the core for which the incident angle on the core-cladding interface equals the critical angle then incident angle of light on the core is called the "Angle of Acceptance. Transmission of light when the incident angle is equal to the acceptance angle If the incident angle is greater than the acceptance angle i.e. $\theta_{i}>\theta_{0}$ then the angle of incidence on the core-cladding interface will be less than the critical angle due to which part of the incident light is transmitted into cladding as shown in the figure below Transmission of light when the incident angle is greater than the acceptance angle If the incident angle is less than the acceptance angle i.e. $\theta_{i}<\theta_{0}$ then the angle of incidence on the core-cladding interface will be greater than the critical angle for which total internal reflection takes place inside the core. As shown in the figure below Transmission of lig
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