## Principle, Construction and Working of Current Carrying Solenoid

Current Carrying Solenoid:

The Solenoid is an artificial magnet which is used for different purposes.

Principle of Solenoid:

The principle of the solenoid is based on the "Ampere Circuital Law" and its magnetic field is raised due to the current carrying a circular loop.

Construction of Solenoid:

The current carrying solenoid is consist of insulated cylindrical material and conducting wire. The conducting wire like copper is wrapped closely around the insulated cylindrical material ( like cardboard, clay, or plastic). The end faces of the conducting wire are connected to the battery.
 Long Current Carrying Solenoid
Working:

When the electric current flow in the solenoid then a field (i.e. Magnetic field) is produced around and within the current carrying solenoid. This magnetic field is produced in solenoid due to circular loops of the solenoid and the direction of the magnetic field is depend upon the direction of the electric current flow in the circular loop. The magnetic field within the current carrying solenoid is uniform and parallel to the axis of the solenoid.

Derivation of the magnetic field due to long current carrying Solenoid:
 Cross Section View of the long Current Carrying Solenoid
Let us consider a very long current carrying solenoid of length $l$ in which $i$ electric current is flowing. Here its diameter is very less as compared to the length of the solenoid.

Now take a closed rectangular path $abcd$ in which the side $ab$ is parallel to the axis of the solenoid and sides $bc$ and $da$ are very long so that the side $cd$ is far from the solenoid and the magnetic field at this side is negligibly small.

Now apply Ampere's circuital law to the rectangular path $abcd$

$\oint \overrightarrow{B}. \overrightarrow{dl}=\mu_{\circ} i' \qquad(1)$

Where $i'$ is the current enclosed by the rectangle.

Let $n$ is the number of turns per unit length of the solenoid. So the number of turns in a length $x$ is = $nx$

The current in each turn is $i$ then the net current $(i')$ enclosed by the rectangle $abcd$ is $nxi$ i.e

$i'=nxi$

Now substitute the value of $i'$ in above equation $(1)$

$\oint \overrightarrow{B}. \overrightarrow{dl}=\mu_{\circ} nxi \qquad(2)$

Now expand the Ampere circuital law for closed rectangular $abcd$-

$\oint \overrightarrow{B}.\overrightarrow{dl}=\int_{a}^{b} \overrightarrow{B}.\overrightarrow{dl}+ \int_{b}^{c} \overrightarrow{B}.\overrightarrow{dl} + \int_{c}^{d} \overrightarrow{B}.\overrightarrow{dl} + \int_{d}^{a} \overrightarrow{B}.\overrightarrow{dl} \qquad(3)$

In above equation the term:

$\int_{b}^{c} \overrightarrow{B}.\overrightarrow{dl}= \int_{d}^{a} \overrightarrow{B}.\overrightarrow{dl}=0$

The above term is zero because along $bc$ and $da$ the magnetic field $\overrightarrow{B}$ and length element $\overrightarrow{dl}$ are perpendicular to each other.

$\int_{c}^{d} \overrightarrow{B}.\overrightarrow{dl}=0$

The above term is zero because the magnetic field $\overrightarrow{B}$ outside the solenoid is negligible due long length of the solenoid.

The above equation $(3)$ can be written by applying the above condition-

$\oint \overrightarrow{B}.\overrightarrow{dl}=\int_{a}^{b} \overrightarrow{B}.\overrightarrow{dl}$

$\oint \overrightarrow{B}.\overrightarrow{dl}= \overrightarrow{B}\int_{a}^{b} \overrightarrow{dl}$

$\oint \overrightarrow{B}.\overrightarrow{dl}= B x \qquad(4)$

Where $x$- Length of $ab$

From equation $(2)$ and equation $(4)$, we get

$Bx=\mu_{\circ}nxl$

$B=\mu_{\circ}nl$

## Comparison of Step Index and Graded Index Fibres

Comparison of Step Index Fibres and Graded Index Fibres(GRIN)→

S.No. Step Index Fibre Graded Index Fibre
1. In a step-index fibre, the refractive index of the core a constant value. In graded-index fibre, the refractive index in the core decreases continuously in a nearly parabolic manner from a maximum value at the centre of the core to a constant value at the core-cladding interface.
2. For a step-index fibre, the variation of refractive index is mathematically expressed as,
$\begin{cases} & \mu(r)=\mu_{1} \qquad 0 < r < a \quad for (core)\\ & \mu(r)=\mu_{2} \qquad r >a \quad for(Cladding)\\ \end{cases} \\ Where \: \mu_{1} > \mu{2}$
Parabolic refractive index variation in GRIN fibre is mathematically expressed as,
$\begin{cases} & \mu^{2}(r)=\mu^{2}_{1} \left[ 1- \left(\frac{r}{\alpha} \right)^{2} \right] \qquad 0 < r < a \quad for (core) \\ & \mu(r)=\mu^{2}_{2} \qquad \qquad \qquad \qquad r > a \quad for (Cladding) \end{cases}$
3. In the step-index fibre, the propagating light rays reflect abruptly from the Core cladding boundary. In graded-index fibre, the propagating light rays bend smoothly as they approach the cladding.
4. for given fibre diameter, the numerical aperture of step-index fibre is large. For the same fibre diameter, the numerical aperture of graded-index fibre is small.
5. In the step-index fibre, there may be some irregularities at the interface between the core and cladding. In the graded-index fibre, there are no such irregularities at the interface between core and cladding.
6. The step-index fibre has higher attenuation. The graded-index fibre has lower attenuation.
7. For a step-index fibre of a given physical size, with a loss of power of the order of $12 \frac{dB}{km}$, the numerical aperture is of the order of $0.2$ to $0.35$. For a graded-index fibre of the same physical size, with an attenuation between $5$ to $10 \frac{dB}{km}$, the numerical aperture tends to run between $0.16$ and $0.2$
8. In step index fibre, the time interval at the output end or pulse dispersion is expressed as,
$\Delta \tau = \frac{\mu_{1} l}{c} \left ( \frac{\mu_{1}}{\mu_{2}} - 1 \right)=\frac{\mu_{1} l}{c} \Delta$
Where $l$ → The length of the fibre.
In a graded index fibre, the time interval at the output end or pulse dispersion is expressed as,
$\Delta \tau = \frac{\mu_{2} l}{2c} \left ( \frac{\mu_{1} - \mu_{2}} {\mu_{2}} \right)^{2}=\frac{\mu_{2} l}{2c} \Delta^{2}$
Where $l$ → The length of the fibre.
9. Pulse dispersion in multimode step-index fibre is large. Pulse dispersion in a graded-index fibre is small.
10. A good quality step-index fibre may have a bandwidth of $50 MHz km$ The equivalent graded-index fibre can have $200$, $400$, or $600 MHz km$ bandwidth.

## Comparison of Single Mode and Multimode Index Fibres

Comparison of Single-Mode Index Fibres and Multimode Index Fibres→

S.No. Single Mode Index Fibre Multimode Index Fibre
1. In single mode index fibre, the diameter of the core is very small and is of the same order as the wavelength of light to be propagated. It is in the range $5\mu m - 10 \mu m$. The Cladding diameter is about $125 \mu m$. In multimode index fibre, the diameter of the core is large. It is in the range $30\mu m - 100 \mu m$. The Cladding diameter is in the range $125 \mu m - 500 \mu m$.
2. The difference in refractive indices of the core and cladding material is very small. The difference in the refractive indices of the core and the cladding materials is large.
3. In single-mode fibre, only a single mode is propagated. In multi-mode fibre, a large number of modes can be propagated.
4. Single mod fibre does require a much more sophisticated light source in order to launch enough light into the tiny core. Multi-mode fibre does not require any sophisticated light source.
5. Single-mode fibre is more expensive but more effective. Multimode fibre is less expensive.
6. The acceptance angle and the size of the acceptance cone of single-mode fibre are small. The acceptance angle and the size of the acceptance cone of multimode fibre are large.
7. The numerical aperture of single-mode fibre is small. The numerical aperture of multimode fibre is large.
8. Single-mode fibre has a very high information-carrying capability. Multimode fibre has low information carrying capability.
9. Single-mode fibre is used when sort distance communication is required. It is used for long-distance communication.
10. Model dispersion in single-mode fibre is almost nil. Model dispersion in multimode fibre is the dominant source of dispersion.
11. Material dispersion in single-mode fibre is low. Material dispersion in multimode fibre is large.
12. When a transmission has a very large bandwidth, single-mode fibre is used Example: Under Sea Cables. When the system bandwidth requirement is low, multimode fibres are used Example: Datalink

## Difference between Fraunhofer and Fresnel diffraction

Difference between Fraunhofer Diffraction and Fresnel Diffraction→

S.No. Fresnel Diffraction Fraunhofer Diffraction
1. The distance between source to slit and slit to screen is finite. The distance between source to slit and slit to screen is infinite.
2. The shape of the incident wavefront on the slit is spherical or cylindrical. The shape of the incident wavefront on the slit is plane.
3. The shape of the incident wavefront on the screen is spherical or cylindrical. The shape of the incident wavefront on the screen is a plane.
4. There is a path difference created between the rays before entering the slit. This path difference depends on the distance between the source and slit. There is not any path difference between the rays before entering the slit.
5. Path difference between the rays forming the diffraction pattern depends on the distance of the slit from the source as well as the screen and the angle of diffraction. Hence the mathematical treatment is complicated. Path difference depends only on the angle of diffraction. Hence the mathematical treatment is comparatively easier.
6. Lenses are not required to observe or perform Fresnel diffraction in the laboratory. Lenses are required to observe or perform Fraunhofer diffraction in the laboratory.