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.

Principle construction and working of potentiometer

Potentiometer- An ideal voltmeter that does not change the original potential difference, needs to have infinite resistance. But a voltmeter cannot be designed to have infinite resistance. The potentiometer is one such instrument that does not draw any current from the circuit and still measures the potential difference. so it behaves as an ideal voltmeter.

"A potentiometer is an instrument. This is used to measure the potential difference between two points of an electric circuit and emf of a cell."

Principle- The principle of the potentiometer depends upon the potential gradient along the wire i.e. "When a constant current flows in a wire then the potential drops per unit length of the wire".

Construction- A potentiometer consists of a long wire $AB$ of uniform cross-section, usually, this wire is $4 m$ to $10 m$ long and it is made of the material having high resistivity and low-temperature coefficient such as manganin or constant. Usually, $1m$ long-separate pieces of wire are fixed on a wooden board. These pieces of wires are parallel to each other and joined in series by thick copper strips. A meter scale is used to measure the position of the jockey on the wire and it is fixed parallel to the wire ( As shown in the figure). The ends $A$ and $B$ are connected to a strong battery, a plug key $(K)$, and a rheostat $(Rh)$. The circuit is called a driving or auxiliary circuit that sends a constant current $i$ through the wire $AB$. Thus the potential gradually drops from $A$ to $B$. This potential drop is measured with the help of a jockey and voltmeter. A jockey is used to make point contact on the wire and it slides on the wire along the length of the wire.
Diagram of Potentiometer
Diagram of Potentiometer
Working- When a constant current flow through a wire of uniform cross-section area and composition, the potential drop across any length of the wire is directly proportional to that length.

If the voltmeter is connected between the end $A$ and Jockey $j$, it reads the potential difference $V$ across the length of the wire $AJ$. So according to ohm's law-

$V=iR$

$V=i \rho \frac{l}{A} \qquad(1)$

for a wire $\rho,i,A$ are constants. So

$\frac{i \: l}{A}=K (Constant)$

Hence equation $(1)$ can be written as

$V=Kl$

$V \propto l$

Metre Bridge OR Slide Wire Bridge

What is Metre Bridge?
It is the simplest practical application of the Wheatstone's bridge that is used to measure an unknown resistance.
Principle: Its working is based on the principle of Wheatstone's Bridge. When the Wheatstone's bridge is balanced

$\frac{P}{Q}=\frac{R}{S}$

Construction: It consists of usually one-meter long manganin wire of uniform cross-section, stretched along a meter scale fixed over a wooden board and with its two ends soldered to two L-shaped thick copper strips $A$ and $C$. Between these two copper strips, another copper strip is fixed so as to provide two gaps $mn$ and $m_{1}n_{1}$. A resistance box (R.B.) is connected in the gap $mn$ and the unknown resistance $S$ is connected in the gap $m_{1}n_{1}$. A cell of emf $E$, Key $(K)$, and rheostat are connected across $AC$. A movable jockey and a galvanometer are connected across the $BD$, as shown in the figure.
Metre Bridge Setup
Metre Bridge Or Slide Wire Bridge

Working: In a Metre bridge, First take out the suitable resistance $R$ from the resistance box after that move the jockey along the wire $AC$ till there is not any deflection in the galvanometer. This is the condition of a balanced Wheatstone's bridge. If $P$ and $Q$ are the resistance of the part $AB$ and $BC$ of the wire, then for the balanced condition of the bridge, we have,

$\frac{P}{Q}=\frac{R}{S} \qquad(1)$

Let us consider:

The total length of the wire $AC=100 \: cm$
Length of the part $AB$ of wire = $l \: cm$
Length of the part $BC$ of wire = $(100-l) \: cm$
Resistance per unit length of the wire = $\sigma$
Resistance of wire of uniform cross-section = $\infty$

$\frac{P}{Q}=\frac{Resistance \: of \: AB}{Resistance \: of \: BC}$

$\frac{P}{Q}=\frac{\sigma l }{\sigma \left( 100-l \right) }$

$\frac{P}{Q}=\frac{ l }{ \left( 100-l \right) } \qquad (2)$

Now substitute the value of equation $(2)$ in equation $(1)$ then we get

$\frac{R}{S}=\frac{ l }{ \left( 100-l \right) } $

$S=\frac{R(100-l)}{l}$

Where

$S$ → Unknown Resistance
$R$ →Standard Resistance

Wheatstone's Bridge

It is an arrangement of four resistance used to determine one of this resistance quickly and accurately in terms of the remaining three resistance.

Objective: To find the unknown resistance with the help of the remaining three resistance.

Principle of Wheatstone Bridge: The principle of Wheatstone bridge is based on the principle of Kirchhoff's Law.

Construction: A Wheatstone bridge consists of four resistance $P$,$Q$,$R$, and $S$. This resistance is connected to form quadrilateral $ABCD$. A battery of EMF $E$ is connected between point $A$ and $C$ and a sensitive galvanometer is connected between point $B$ and $D$ Which is shown in the figure below.
Wheatstone's Bridge
Diagram of Wheatstone's Bridge
Working: To find the unknown resistance $S$, The resistance $R$ is to be adjusted like there is no deflection in the galvanometer. which means that there is not any flow of current in the arm $BD$. This condition is called "Balanced Wheatstone bridge" i.e

$\frac{P}{Q}=\frac{R}{S}$

Derivation of Balanced Condition of Wheatstone's Bridge: In accordance with Kirchhoff's first law, the currents through various branches are shown in the figure above.

For Close Loop $ABDA$

$0=P \: I_{1} - R \: I_{2} +G \: I_{g} \qquad(1)$

For Close Loop $CBDC$

$0=Q \left( I_{1} - I_{g} \right) -S \left( I_{2} + I_{g} \right) - G\: I_{g} \qquad(2)$

For Balanced Wheatstone Bridge: $I_{g}=0$

SO from equation $(1)$ and equation $(2)$

$0=P \: I_{1} - R \: I_{2} $

$P \: I_{1} = R \: I_{2} \qquad(3)$

$0=Q I_{1} -S I_{2}$

$Q I_{1} = S I_{2} \qquad(4)$

Now divide the equation $(4)$ and equation $(3)$, then we get

$\frac{P}{Q}=\frac{R}{S}$

Kirchhoff's laws for an electric circuits

Kirchhoff's laws: Kirchhoff had given two laws for electric circuits i.e.
  1. Kirchhoff's Current Law or Junction Law

  2. Kirchhoff's Voltage Law or Loop Law
  1. Kirchhoff's Current Law or Junction Law: Kirchhoff's current law state that

    The algebraic sum of all the currents at the junction in any electric circuit is always zero.

    $\sum_{1}^{n}{i_{n}}=0$

    Sign Connection: While applying the KCL, the current moving toward the junction is taken as positive while the current moving away from the junction is taken as negative.
    Flow of Current in a Junction
    The flow of Current in a junction
    So from figure,the current $i_{1}$,$i_{2}$,$i_{5}$ is going toward the junction and the current $i_{3}$,$i_{4}$, So

    $\sum{i}= i_{1}+i_{2}+(-i_{3})+(-i_{4})+i_{5}$

    According to KCL $\sum{i}= 0$, Now the above equation can be written as

    $i_{1}+i_{2}+(-i_{3})+(-i_{4})+i_{5}=0 \qquad$

    $i_{1}+i_{2}+i_{5}=i_{3}+i_{4}$

    Thus, the sum of current going towards the junction is equal to the sum of current going away from the junction.

    In other words, at any junction, neither the charge accumulates nor the charge is removed. So this law represents conservation of charge.


  2. Kirchhoff's Voltage Law or Loop Law: Kirchhoff's voltage law state that:

    The algebraic sum of all the voltage or emf in any closed loop of an electric circuit is always zero.

    $\sum_{1}^{n}{E_{n}}=0$

    This means that the algebraic sum of all the emf applied in any closed loop is always equal to the algebraic sum of the product of current and resistance in the closed loop.

    $\sum{E}=\sum {i.R}$

    Sign Connection: While applying this law, a product of current and resistance is taken as positive when we traverse in the direction of the conventional current and the emf is taken positively when we traverse from negative to the positive electrode through the electrolyte.
    Distribution of current in a loop of circuit
    Distribution of Current in a Loop of Circuit
    So from the figure:

    For Mesh $(1)$

    $E_{1}-E_{2}=i_{1}R_{1}-i_{2}R_{2} \qquad(1)$

    For Mesh $(2)$

    $E_{2}=i_{2}R_{2}+\left( i_{1}+i_{2} \right)R_{3} \qquad(2)$

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