Combination of cell in the circuit

A.) Combination of cells when emf of cells are same: There are three types of combinations of cells in the circuit

1.) Series Combination of Cells

2.) Parallel Combination of Cells

3.) Mixed Combination of Cells

1.) Series Combination of Cells: Let us consider that the $n$ - cells having emf (electromotive force) $E$ and internal resistance $r$ are connected in series with external resistance $R$. Then from the figure given below

The total emf of the $n$ - cell = $nE$

The total internal resistance of the $n$ - cell = $nr$

The total resistance of the circuit = $nr+R$

The total current in the circuit

$i=\frac{Total \: emf \: of \: the \: n - series \: cell}{Total \: resistance \: of \: the \: circuit}$

$i=\frac{nE}{nr+R}$

2.) Parallel Combination of Cells: Let us consider that the $n$ - cells having emf (electromotive force) $E$ and internal resistance $r$ are connected in parallel with external resistance $R$. Then from the figure given below

The total emf of the $n$ - cell = $E$

The total internal resistance of the $n$ - cell

$\frac{1}{r_{eq}} = \frac{1}{r}+ \frac{1}{r}+.........n \: times$

$\frac{1}{r_{eq}}=\frac{n}{r}$

$r_{eq}=\frac{r}{n}$

The total resistance of the circuit = $\frac{r}{n}+R$

The total current in the circuit

$i=\frac{Total \: emf \: of \: the \: n - parallel \: cell}{Total \: resistance \: of \: the \: circuit}$

$i=\frac{E}{\frac{r}{n}+R}$

$i=\frac{E}{\frac{r+nR}{n}}$

$i=\frac{nE}{r+nR}$

3.) Mixed Combination of Cells: Let us consider that the $n$ - cells having emf (electromotive force) $E$ and internal resistance $r$ are connected in series in each row of $m$ parallel rows with external resistance $R$. Then from the figure given below

The total emf of the $n$ - cell in each row of $m$ parallel rows of the cells = $nE$

The internal resistance of the $n$ - cell in each row = $nr$

The total internal resistance of the $n$ - cell in each of $m$ parallel rows of the cells = $nr$

$\frac{1}{r_{eq}} = \frac{1}{nr}+ \frac{1}{nr}+.........m \: times$

$\frac{1}{r_{eq}}=\frac{m}{nr}$

$r_{eq}=\frac{nr}{m}$

The total resistance of the circuit = $\frac{nr}{m}+R$

The total current in the circuit

$i=\frac{Total \: emf \: of \: the \: cell}{Total \: resistance \: of \: the \: circuit}$

$i=\frac{nE}{\frac{nr}{m}+R}$

$i=\frac{nE}{\frac{nr+mR}{m}}$

$i=\frac{mnE}{nr+mR}$

It is clear from the above equation that for the value of $i$ to be maximum, the value of $(nr+mR)$ should be minimum. Now,

$nr+mR= \left[ \sqrt{nr}-\sqrt{mr} \right]^{2}+2 \sqrt{mnRr}$

Therefore, for $(nr+mR)$ to be minimum, the quantity $\left[ \sqrt{nr}-\sqrt{mr} \right]^{2}$ should be minimum. So

$\left[ \sqrt{nr}-\sqrt{mr} \right]^{2} = 0$

$ \sqrt{nr}-\sqrt{mr} = 0$

$ \sqrt{nr} = \sqrt{mr} $

$nr=mR$

$R=\frac{nr}{m}$

Here, $\frac{nr}{m}$ is the total resistance of the cells.

Thus, When the total internal resistance of the cells are equal to the external resistance then the total current in the external circuit will be maximum in the mixed combination of cells.

Read More

Relation between electromotive force (E), internal resistance (r) and potential difference (V) in a circuit

Relation between electromotive force $(E)$, internal resistance $(r)$ and potential difference $(V)$:

Let us consider:

The cell having electro-motive force = $E$

The cell having internal resistance = $r$

The external resistance of the circuit = $R$

The potential difference between the external resistance of the circuit = $V$

The current in circuit = $i$

So, The emf of the cell from the given circuit in the figure above

$E=iR+ir$

$E = V+ir$

$V=E-ir$

Read More

Resolving Power of Optical Instrument | Rayleigh Criterion of Resolution

Resolving power of an optical instrument:

The ability of an optical instrument to just resolve the images of two closely spaced objects is called its resolving power.

Limit of Resolution:

The smallest distance between two closely spaced objects that can be seen as separated or just separated from each other through an optical instrument is known as the limit of resolution of that optical instrument.

Rayleigh Criterion:

Rayleigh criterion describes the separation between the two objects or wavelengths (i.e. resolving power) by the resultant intensity distribution of objects and wavelengths. According to Rayleigh's criterion, there are the following cases:

Case:1 If two point sources have very small angular separation, then central or principal maxima in their diffraction patterns will overlap to a large extent and resultant intensity shows uniform variation. As shown in the figure below. In this case, the two objects or wavelengths can not be distinguished or unresolved.

Case:2 If two point sources have very large angular separation then the central or principal maxima are widely separated and the resultant intensity shows two widely separated peaks. As shown in the figure below. In this case, the objects or wavelengths are resolved well.

Case:3 If the central or principal maxima in the diffraction pattern of one object or wavelength coincide with the first minima in the diffraction pattern of the other objects or wavelength then the resultant intensity shows a small dip. As shown in the figure below. In this case, the objects or wavelengths are seen to be just separate or just resolved.

Read More

Attenuation of optical signal in optical fibre

Attenuation (or Loss) of optical signal in optical Fibre:

The difference in the power of the input optical signals and output optical output signals in optical fibre is known as attenuation in the optical fibre. It is measured in decibels per kilometer $(\frac{dB}{Km})$ and caused by the absorption and scattering of the optical signal in optical fibre.

The optical signal strength is reduced when the signal travels in optical fibre over a long distance. The expression for the attenuation of optical signal in optical fibre:

$\alpha = - \frac{10}{x(Km)} log \left[ \frac{P_{x}}{P_{\circ}} \right]$

Where
$P_{x} \rightarrow $ Power of optical signal at a position $x$ from origin
$P_{\circ} \rightarrow $ Power of optical signal at origin

The Losses in optical fibre are wavelength-dependent and the attenuation factor depends on the fibre material and manufacturing tolerance.

Types of Attenuation or Loss

A.) Absorption Attenuation or Loss
B.) Scattering Attenuation or Loss
C.) Bending Attenuation or Loss

A.) Absorption Attenuation or Loss:

The absorption of optical signals in optical fibre depends on the amount of the material in optical fibre. There are two types of absorption on optical fibre:

1.) Intrinsic Material Absorption
2.) Extrinsic Material Absorption

1.) Intrinsic Material Absorption: It is a fundamental minimum loss due to absorption when the optical signal passes through the optical fibre. This absorption occurs due to material in optical fibre with no impurities.

2.) Extrinsic Material Absorption: It is a loss due to absorption when the optical signal passes through the optical fibre. This absorption occurs due to a material having impurities ( such as $Fe^{+2}, Cu^{+2}, Cr^{+3}, and \: OH^{-}$ ion from water dissolved in glass) in the optical fibre material.

B.) Scattering Attenuation or Loss:

When the optical signal interacts with a particle then the energy of the optical signal is reduced and it goes in another direction. In scattering, the optical signal is not absorbed but it goes to any other direction which also causes the loss of the optical signal. Scattering is the loss of optical signal due to imperfections in the optical fibre i.e. the basic structure of the optical fibre. There are two types of scattering that occur in the optical fibre i.e.

1.) Linear Scattering:
2.) Nonlinear Scattering

1.) Linear Scattering: The amount of power of the optical signal transferred from an optical wave is proportional to power. There is no change in the frequency of the optical signal.

Types of Linear Scattering

i.) Rayleigh scattering in optical fibre
ii.) Mie scattering in optical fibre

i.) Rayleigh scattering in optical fibre: When the optical signal interacts with the lattice of the core it causes the scattering because the size of the lattice is smaller than the wavelength of the incident optical signal. This interaction is also known as elastic scattering.

In optical fibres, This scattering controls the intrinsic loss mechanism in the low-absorption between absorption tails of the ultraviolet and infrared. This is caused by random heterogeneity in the material of the core lattice that leads to changes in the refractive index. These changes cause attenuation in the optical signal by scattering in optical fibre.

ii.) Mie scattering in optical fibre: When the optical signal interacts with the lattice of the core that causes the scattering because in this scattering the size of a lattice is comparable to the wavelength of the incident optical signal. This interaction is also known as elastic scattering.

In optical fibres, This scattering occurs when the optical signal interacts with inhomogeneities in the fibre core or cladding that are larger than the wavelength of the incident optical signal.

2.) Nonlinear scattering: It is an inelastic scattering that occurs when an optical signal interacts with a material in a non-linear manner. Non-linear scattering occurs, when the intensity of the incident optical signal is very high to change the refractive index of material or when the optical signal interacts with molecules or particles that have non-linear optical properties.

In non-linear scattering, the intensity of the scattered optical signal is not proportional to the intensity of the incident optical signal, and the scattered optical signal can be polarized in different planes than the incident optical signal.

In other words,

In non-linear scattering the wavelength, frequency, or phase of scattered optical signal is different than the incident optical signal. The energy and momentum of the scattered optical signal are not conserved, hence, the scattered optical signal has different characteristics than the incident optical signal.

Types of Non-linear Scattering:

i.) Raman Scattering
ii.) Brillouin Scattering
iii.) Self-phase modulation (SPM)
iv.) Cross-phase modulation (XPM)
v.) Four-wave mixing (FWM)

i.) Raman Scattering: It is an inelastic scattering that occurs when a photon of optical signal interacts with molecules of vibrational modes in the material of the core then this photon excites the molecule to a virtual state before being re-emitted in a different direction with a different energy and frequency.

The frequency shift of the scattered optical signal is directly related to the molecule's vibrational energy, which is a characteristic property of the material.

Stimulated Raman scattering $(SRS)$ is an optical process (i.e. nonlinear process ) that enhances scattering by stimulating the molecules of vibrational modes. In SRS, a pump beam of high intensity is used to amplify the Raman signal which leads to a strong and more easily detectable output optical signal.

ii.) Brillouin Scattering: It is an inelastic scattering that occurs when an optical signal interacts with a material, usually a solid or a liquid, causing the material to vibrate.

It can also be said to be a scattering that occurs due to the interaction of an incident optical signal with the acoustic wave in a material.

The vibrating material scatters the optical signal, shifting its frequency by an amount proportional to the frequency of the vibration.

The scattering of the light by the vibrating material is called the Brillouin scattering. This scattering can be either stimulated or spontaneous. It depends on the material that is excited by an external stimulus or its own oscillations.

Stimulated Brillouin scattering (SBS) is used to control the optical signals in optical fibers and other waveguides. When the frequency of the incident optical signal is tuned to the frequency of the vibrational modes of the fiber then the scattered optical signal is shifted to a new frequency which allows for attenuation or amplification of the optical signal.

Brillouin scattering has various applications in materials science. It is used to study the mechanical properties of materials and to determine the stress and strain in solid materials.

iii.) Self-phase modulation (SPM): It is a non-linear effect that occurs when the optical signal of high intensity travels through an optical fiber. The intensity of the optical signal causes a change in the refractive index of the optical fiber that causes a phase shift in the optical signal.

This phase shift can cause the optical signal to spread out in time and frequency which leads to distortion of the transmitted signal.

In other words, The optical signal pulses change their spectrum due to their own intensity from an induced varying refractive index of the medium.

iv.) Cross-phase modulation (XPM): It is a non-linear effect that occurs when two optical signals, typically from two different channels, interact with each other in an optical fiber.

The interaction between the optical signals causes a change in the refractive index of the fiber, that causes a phase shift in one of the optical signals.

This phase shift can distort the optical signal which can lead to crosstalk between different channels in a fiber optic communication system.

v.) Four-wave mixing (FWM): It is a non-linear effect that occurs when two or more incident optical signals interact with each other in a non-linear medium and produce a new wave with a different frequency and phase.

It is used for wavelength conversion of optical signals and signal processing in optical fibers.

C.) Bending Attenuation or Loss: It is caused by the bending of optical fibre or physical stress on the fibre.

Types of Bend Loss

a.) Microbend Loss
b.) Macrobend Loss

a.) Microbend Loss: It is caused by small discountinuities or imperfections in the optical fibre. Microbend loss increses due uneven coating applications and improper cabling procedures. The external force is also a source of micro bending.

b.) Macrobend Loss: It is caused when the fibre bend's radius of curvature is larger than the fibre diametre. These bends are a great source of loss when the radius of curvature is less than several centimetres.

Read More

Principle of optical fibre communication

Optical Fibre Communication:

Optical fibre communication is a method of communication in which an optical wave passes through optical fibre by the total internal reflection principle. This optical signal consists of the electrical signal (Also known as information) and the laser beam i.e. a carrier wave. The optical fibre is used as a waveguide or transmission medium in optical fibre communication.

Principle of optical fibre communication:

In the optical fibre communication principle, the information (such as voice) is first converted into an electrical signal. Then it is modulated onto the laser beam (Also known as a carrier wave). In the modulation process, The electrical signal is superimposed onto the laser beam and the frequency of laser light changes with the frequency of the electrical signal. The optical fibre communication uses the pulse code modulation (PCM) for transmitting the optical signal. Now modulated optical signal passes through the optical fibre or waveguide, by the principle of total internal reflection, from the transmitter to the receiver.

The receiver receives the optical signal and demodulates it. In the demodulation process, the detector detects the signal and removes the carrier wave from the electrical signal. So the original information is received at the receiver end as sent by the transmitter.

Optical fibre communication follows the principle of total internal reflection. When the light is incident at a certain angle into the core of the optical fibre, total internal reflection occurs within the optical fibre causing long-distance transmission.

Light in the optical fibre can be classified into two types: meridional rays, which travel along the meridional plane, and oblique rays, which propagate at an angle to the fibre axis.

Read More

Description dielectric materials and their types

Dielectric Materials:

Materials that do not allow current to flow through them are called insulators or dielectrics. Dielectric materials are capable of storing electric energy. Dielectric materials do not have free electrons ( in the case of ideal dielectric) because electrons are tightly bound with the nucleus so the conductivity of the dielectric is poor and for an ideal dielectric, it is zero.

When a dielectric is placed in an external electric field, atoms or molecules of the dielectric material are polarised due to the creation of an electric dipole in the atoms or molecules, and the internal field is set up in the dielectric material which opposes the external applied electric field, thereby reducing the net electric field and hence the electric potential difference. If these dielectrics are placed between plates of a capacitor, the potential difference will be reduced without affecting the charge on the plates.

According to the band theory of solids,

A dielectric is a material in which the energy band gap between balance and conduction band is more than three electron volt.

Examples: Glass, Plastic, Mica, Rubber, Wood, Turpentine oil etc.

Types of Dielectric materials:

Dielectric materials are of two types :

1.) Non-polar dielectrics
2.) Polar dielectric

1.) Non-polar dielectrics:

In non-polar dielectric materials, the molecules that are usually diatomic and composed of the same type of two atoms have a symmetrical structure that is the positive nuclei are surrounded by a symmetrically distributed negative electron cloud. The center of gravity of positive and negative charge distribution coincide and so the molecules are electrically neutral and have zero electric dipole moment.

Examples: $H_{2}$, $O_{2}$, $CO_{2}$, $CCl_{4}$, $C_{6}H_{6}$, $C_{6}H_{12}$, $CS_{2}$ etc.

Polarisation of the non-polar dielectric materials:

When a non-polar dielectric is placed in an external electric field the positive charge of the nucleus and negative charge of the electron cloud experience electric force which causes a displacement between the positive and negative parts of the molecule from their equilibrium position in opposite directions. The distance moved is very small $(10^{-10})$ because the displacement is restricted by storing force which increases with the increase of displacement. Therefore the centre of gravity of these positive and negative charges no longer coincide and molecules are said to be polarized. The molecules does acquire an induced electric dipole moment and aligned in the direction of external field. The induced dipole moment and the polarization disappear when an electric field is removed.

2.) Polar Dielectrics:

In polar dielectric materials, the molecules which are normally composed of two or more different atoms have permanent dipole moments because the center of gravity of these positive charges and that of the negative charges in a molecule are permanently separated by a finite but small distance. This is due to the asymmetric shape of the molecule. Thus each molecule in the polar dielectric material behaves as a dipole having a permanent dipole moment. Normally these molecules are in polar dielectrics and randomly arranged such that the net dipole moment is zero and the material acts as a neutral one.

Examples: $H_{2}O$, $CHCl_{3}$, $C_{6}H_{5}Cl$, $C_{6}H_{5}NO_{2}$, $C_{2}H_{5}OH$, $NH_{3}$, $HCl$, $CO$,etc.

Polarization of the polar dielectric materials:

when polar dielectric is placed in an external electric field the molecular dipole tense to align themselves in the direction of the field and acquire a considerable amount of dipole moment. this the dielectric acid to be polarised.

In the $HCL$ molecule, the electron of the $H$ atom lies more toward the $Cl$ atom. The $H$ end of the $HCl$ molecule is positive and the $Cl$ end is negative. The molecule is therefore a dipole having dipole moment $\overrightarrow{p}$ direct from $Cl$ atom to $H$ atom.

Hence, the polarization of non-polar dielectric material is the displacement of positive and negative charge, and in the case of polar dielectric material, the polarization is the orientation of molecular dipole moment under the action of the electric field to which they are subject.

Read More

Difference between Potentiometer and Voltmeter

There are the following differences between a potentiometer and a voltmeter given below:

Potentiometer:

1.) It is based on null method.

2.) It gives an accurate value of emf.

3.) While measuring emf, it does not draw any current from the cell.

4.) Resistance of potentiometer wire becomes infinite while measuring emf.

5.) It can be used for various experimental purposes.

6.) It can not be taken conveniently from one place to another place.

Voltmeter:

1.) It is based on the deflection method.

2.) It does not give an accurate value of emf.

3.) While measuring emf, it draws some current from the cell. Hence it reads slightly less than the actual emf.

4.) The resistance of the voltmeter is high enough but not infinite.

5.) It can be used to measure potential differences only.

6.) It can be conveniently taken from one place to another place.

Read More

Principle Construction, Working and Angular Magnification of Simple Microscope

Principle of Simple Microscope:

The principle of the simple microscope is based on the magnification of an image by using a simple convex lens.

Construction:

A simple microscope consists of one convergent lens only. The object is placed between the lens and its focal length, and the eye is placed just behind the lens. Then the eye sees a magnified, erect, and virtual image on the same side as the object at the least distance of distinct vision $(D)$ from the eye, and the image is then seen most distinctly.

Working:

If the small object $ab$ is placed between a lens $O$ and its first focus $f$ then Its magnified virtual image $a_{1}b_{1}$ is formed at a distance $D$ from the lens. Since the eye is just behind the lens, the distance of image $a_{1}b_{1}$ from the eye is also $D$.

Angular Magnification Or Magnifying Power($M$):

The ratio of the angle subtended by the image at the eye ($\beta$) to the angle subtended by the object at the eye when placed at the least distance of distinct vision ($\alpha$) is called the angular magnification or magnifying power.

$M= \frac{Angle \: subtended \: by \: the \: image \: at \: the \: eye \: (\beta)}{Angle \: subtended \: by \: the \: object \: at \: the \: eye \: when \\ placed \: at \: least \: distance \: of \: distinct \: vision \: (\alpha)}$

$M=\frac{\beta}{\alpha} \approx \frac{tan \beta}{tan \alpha} \quad (1)$

From figure

$tan \beta = \frac{ab}{oa} $

$tan \alpha = \frac{a_{1}b_{2}}{a_{1}o}$

Here $a_{1}b_{2} = ab$

$tan \alpha = \frac{ab}{a_{1}o}$

Now substitute these values in equation $(1)$, then

$M=\frac{\frac{ab}{ao}}{\frac{ab}{a_{1}o}}$

$M=\frac{a_{1}o}{ao}$

Here $ao = u$ (Distance between object and optical center of the lens) and $a_{1}o = D$ (Least Distance of distinct vision), then the above equation can be written as

$M=\frac{D}{u} \qquad(2)$

We know that the lens formula $\frac{1}{v}-\frac{1}{u} = \frac{1}{f}$

Now put
$v=-D$ (The image $a'b'$ is being formed at a distance $D$ from lens)
$u=-u$

$\frac{1}{-D}-\frac{1}{-u} = \frac{1}{f}$

Multiply $D$ in the above equation

$-\frac{D}{D}-\frac{D}{-u} = \frac{D}{f}$

$-1-\frac{D}{-u} = \frac{D}{f}$

$\frac{D}{u} =1 + \frac{D}{f} \qquad(3)$

From equation $(2)$ and equation $(3)$, then

$M=1 + \frac{D}{f} $

If eye is kept at distance $d$ from lens then $v=-(D-d)$, and the magnifying power will be

$M=1+\frac{D-d}{f}$

To see with a relaxed eye, the image $a'b'$ should be formed at infinity. In this case, the object $ab$ will be at the focus of the lens, i.e. $u=f$ then magnifying power

$M= \frac{D}{f} $

Read More

Origin of Biomedical Signals

The biomedical signals differ from other signals only in terms of the application — signals that are used in the biomedical field. As such, biomedical signals are produced from a variety of sources. The following is a brief description of these sources:

1. Bioelectric signals: The bioelectric signal is unique to biomedical systems. It is produced by nerve cells and muscle cells. It is produced due to the membrane potential, which under certain conditions may be excited to generate an action potential. In single-cell measurements, the specific microelectrodes are used as sensors, and the action potential itself is considered as the biomedical signal. In more gross measurements, the surface electrodes are used as sensors, and the electric field generated by the action of many cells, distributed in the electrode’s vicinity, constitutes the bioelectric signal. Bioelectric signals are probably the foremost biosignals. The fact that most biosystems use excitable cells makes it possible, to use biosignals to study and monitor the main functions of the systems. The electric field propagates through the biological medium, and thus the potential may be acquired at relatively convenient locations on the surface, eliminating the need to invade the system. The bioelectric signal is acquired by a relatively simple transducer. A transducer is required in the field of biomedical because the electric conduction in the biomedical medium is executed through ions, while the conduction in the measurement system is executed through electrons. All these lead to the fact that the bioelectric signal is broadly used in most of the fields of biomedicine.

2. Bioimpedance signals: The impedance of the tissue contains important information related to its composition, blood volume, blood distribution, endocrine activity, autonomic nervous system activity, and many more. The bioimpedance signal is usually generated by injecting into the tissue under test sinusoidal currents (frequency range of $50 kHz–1 MHz$, with low current densities of the order of $20–20 mA$). The electrode polarization problems are minimized by choosing the frequency range and the low current densities are selected to prevent tissue damage mainly due to heating effects. Bioimpedance measurements are usually performed with four electrodes. Two electrodes (known as source electrodes) are used to inject the current into the tissue and these electrodes are connected to a current source. Remaining two electrodes (known as measuring electrodes) are placed on the tissue under investigation and used to measure the voltage drop generated by the current and the tissue impedance.

3. Bioacoustic signals: Many biomedical phenomena create acoustic noise. The measurement of this acoustic noise gives information about the underlying phenomenon. The flow of blood in the heart (i.e through the heart’s valves, or through blood vessels) generates typical acoustic noise. The flow of air through the upper and lower airways and in the lungs generates acoustic sounds. These sounds are called coughs, snores, and chest and lung sounds. These sounds are used extensively in medicine. Sounds are also produced in the digestive tract and in the joints. It also has been observed that the contracting muscle generates an acoustic noise or muscle noise. Since the acoustic energy propagates through the biological medium, the bioacoustic signal may be conveniently acquired on the surface, using acoustic transducers (microphones or accelerometers).

4. Biomagnetic signals: Many organs, such as the brain, heart, and lungs, produce extremely weak magnetic fields. The measurements of these fields provide information but are not included in other biosignals (such as bioelectric signals). Due to the low level of the magnetic fields to be measured, biomagnetic signals are usually of a very low signal-to-noise ratio. Extreme precautions must be taken in designing or developing the acquisition system of these signals.

5. Biomechanical signals: The term biomechanical signals includes all signals used in the biomedicine fields that originate from some mechanical function of the biological system. These signals include motion and displacement signals, pressure and tension signals, flow signals, and others. The measurement of bio-mechanical signals requires a variety of transducers, not always simple and inexpensive. The mechanical phenomenon does not propagate in biomedical signals, as do the electric, magnetic, and acoustic fields. Hence the measurement usually has to be performed at the exact site. This very frequently complex the measurement and forces it to be an invasive one.

6. Biochemical signals: The chemical measurements from the living tissue or from samples analyzed in the clinical laboratory produce biochemical signals. Measuring the concentration of various ions inside and around a cell using specific ion electrodes. It is an example of such a signal. Partial pressures of oxygen (pO2) and carbon dioxide (pCO2) in the blood or respiratory system are other examples. Biochemical signals are often very low-frequency signals. Mostly, biochemical signals are actually DC signals.

7. Biooptical signals: Bio-optical signals are the result of optical functions of the biological system, occurring naturally or induced by the measurement. Blood oxygenation may be analyzed by measuring the transmitted and backscattered light from a tissue ( i.e.in vivo and in vitro) in several wavelengths. Important information about the fetus may be acquired by measuring the fluorescence characteristics of the amniotic fluid. Analysis of the Heart output may be performed by the dye dilution method, which requires the observation of the appearance of recirculated dye in the bloodstream. The development of fiberoptic technology has opened vast applications of bio-optical signals.

Read More

Principle Construction, Working and Angular Magnification of Compound Microscope

Principle: The principle of the compound microscope is based on the magnification of an image by using two lenses.

Construction: A compound microscope consists of two convergent lenses (i.e. objective lens $O$ and eye-piece lens $e$) placed coaxially in a double tube system. The objective lens is an achromatic convergent lens system of short focal length and short aperture. The other eye-piece lens $e$ is also an achromatic convergent lens system of large focal length and large aperture. The observation is taken through the eye-piece lens by the observer. The eye-piece lens is fitted outer side of a movable tube and the inner side connects with a non-movable tube in which the objective lens is fitted on another side of the non-movable tube. The separation between the objective or eye-piece lens can be changed by an arrangement, this is known as rack and pinion arrangement.

Working: Suppose a small object $ab$ is placed slightly away from the first focus $f_{\circ}$ of the objective lens which forms a real, inverted, and magnified image $a_{1}b_{1}$. Now adjust the eye-piece lens by moving like this, that the image $a_{1}b_{1}$ lies in between the optical center and the second focal length $f_{e}$ of the eye-piece lens. This image $a_{1}b_{1}$ works as an object for the eye-piece lens which forms a magnified, virtual, and final image $a_{2}b_{2}$. The final image $a_{2}b_{2}$ is generally formed at the least distance $D$ of distinct vision, although it can be formed anywhere between this position and infinity.

Angular Magnification Or Magnifying Power($M$):

The angular magnification or magnifying power can be defined as the ratio of the angle subtended by the image at the eye ($\beta$) to the angle subtended by the object at the eye when placed at least distance of distinct vision ($\alpha$)

$M= \frac{Angle \: subtended \: by \: the \: image \: at \: the \: eye \: (\beta)}{Angle \: subtended \: by \: the \: object \: at \: the \: eye \: when \\ placed \: at \: least \: distance \: of \: distinct \: vision \: (\alpha)}$

$M=\frac{\beta}{\alpha} \approx \frac{tan \beta}{tan \alpha} \quad (1)$

From figure

$tan \beta = \frac{a_{2}b_{2}}{a_{2} e} $

$tan \alpha = \frac{a_{2}a_{3}}{a_{2}e}$

Now subtitute these values in equation $(1)$, then

$M=\frac{\frac{a_{2}b_{2}}{a_{2} e}}{\frac{a_{2}a_{3}}{a_{2}e}}$

$M=\frac{a_{2}b_{2}}{a_{2}a_{3}}$

Here $a_{2}a_{3} = ab$

So the above equation can be written as

$M=\frac{a_{2}b_{2}}{ab}$

$M=\frac{a_{2}b_{2}}{ab} \frac{a_{1}b_{1}}{a_{1}b_{1}}$

$M=\frac{a_{2}b_{2}}{a_{1}b_{1}} \frac{a_{1}b_{1}}{ab}$

Here $m_{e}=\frac{a_{2}b_{2}}{a_{1}b_{1}}$ and $m_{\circ}= \frac{a_{1}b_{1}}{ab}$

Now substitute the values of $m_{e}$ and $m_{\circ}$ in the above equation

$M=m_{e} \times m_{\circ} \qquad(1)$

Where
$m_{e} \rightarrow$ The linear magnification produced by eye-piece lens system
$m_{\circ} \rightarrow$ The linear magnification produced by the object lens system

So now again from the figure

The linear magnification produced by object lens system $m_{\circ} = -\frac{v_{\circ}}{u_{\circ}}$

The linear magnification produced by eye-piece lens system $m_{e} = \frac{D}{u_{e}}$

Substitute the value of $m_{\circ}$ and $m_{e}$ in equation $(1)$

$M= -\frac{v_{\circ}}{u_{\circ}} \left( \frac{D}{u_{e}} \right) \qquad(2)$

Adjustment of a Compound Microscope:

1.) Adjustment for Clear Vision: In this final image an object is formed at least a distance of distinct vision $D$. For this configuration,

On substitution $u=-u_{e}$, $v=-D$ and $f=f_{e}$ in the lens formula for eyepiece lens

$\frac{1}{-D}+\frac{1}{u_{e}}=\frac{1}{f_{e}}$

$\frac{1}{u_{e}}=\frac{1}{f_{e}} + \frac{1}{D}$

$\frac{D}{u_{e}}= \left( \frac{D}{f_{e}} + 1 \right)$

Now substitute the value of $\frac{D}{u_{e}}$ in equation $(2)$

$M= -\frac{v_{\circ}}{u_{\circ}} \left(1+ \frac{D}{f_{e}} \right) $

The length of the microscope tube in this setup

$L=$ Distance between the object and the eye-piece lenses

$L= v_{\circ} + |u_{e}|$

2.) Adjustment for Relaxed Eye: In this configuration, the final image of an object is formed in a relaxed eye position i.e. at $\infty$. In this setup, the eye-piece lens system is moved back until the image of object $ab$, formed by object lens,i.e., $a_{1}b_{1}$ fall at (coincide with)second focus $f'_{e}$ of the eye-piece lens system. Mathematically this situation comes when $u_{e} = f_{e}$

Thus, the magnifying power in this position,

$M=-\frac{v_{\circ}}{u_{\circ}} \left( \frac{D}{f_{e}} \right)$

For this set the length of the microscope tube,

$L=v_{\circ}+f_{e}$

Read More