UNIT 1

Oscillation, Ultrasonic And Dielectric Materials

1 Question: What is the difference between forced and damped oscillator?

Solution: Damped oscillations: are those oscillations in which amplitude diminishes with time and finally the oscillations stop. In our day-to-day life, we come across many examples where we initiate oscillations by applying one time force or jerk.

Example: Dropping a swing from a height, pulling a spring once and leaving it to oscillate, striking a tuning fork on rubber pad, and so on. In all such cases, oscillations start and die out after some time due to friction. Hence these are called damped oscillations.

Damped oscillations occur at natural frequency of the oscillating system.

Forced oscillations: Those oscillations in which oscillations continue upto the desired point of time with constant amplitude, by application of external periodical force.

Example: vibrations caused in a mobile phone, Oscillations caused by application of alternating current or voltage, turning a wheel manually are its examples.

In case of forced oscillations, frequency of oscillation is determined by frequency of external source i.e. AC generator or oscillator.

2 Question: Derive Differential Equation for Forced Oscillator and also Discuss how amplitude of oscillations vary with damping constant?

Consider a body of mass m attached to one end of the spring whose other end is fixed rigidly. Let an external periodic force Fe(t)=F0cosωet is applied to the body to displace it from its mean position.

Fo= Amplitude of external force

ωe= Angular frequency of external periodic force

With the external force Fe(t) other forces are also acting on oscillators we named it as f1 and f2.

f1= Restoring Force = -kx where k is spring constant and x is displacement. You are thinking of this negative sign. Here Negative sign indicates restoring force acting in opposite direction of the displacement.

f2 = Frictional Force = -bv = -b = -b

Total Force F= f1 +  f2.+ Fe(t) 

F=-kx –b ( ) + F0cosωet               …..(1)    

We know by newton’s law F=ma =m (d2x/ d2y) = m ……(2)

Note- Derivative express as dot on it as written above

m (d2x/ d2y) = -kx –b ( ) + F0cosωet 

m (d2x/ d2y) +kx +b ( ) = F0cosωet

Dividing both sides by m, we get

(d2x/ d2y) +x +( ) = cos ωet

Or + +x = cos ωet  ……(3)

Which is the Differential Equation of Forced Oscillator.

The solution of above (3) eq is given by

x= Xm cos (ωet- ϕ)

Xm=amplitude of forced oscillator and it is given by 

Xm=F0/{m2(ω2-ωe2)2+b2ωe}1/2 …….(4)

=bωe/m(ω2-ωe2)

If damping constant is very small then b2 can be neglected so

Xm= F0/ m(ω2-ωe2)   …….(5)

Amplitude of forced oscillator is maximum when ω=ωe where ω is natural frequency (from eq 4)
Xm= F0/ bωe = F0/ bω

In the absence of damping constant b the maximum amplitude of the forced oscillator at ω=ωe   So Xm= ∞

In figure f0 is frequency = ω=ωe   in case of resonance as predicted in figure we obtain maximum amplitude at f0= = ω/2=ωe/2. 

3 Question: The bus begins to make a loud rattling sound at a certain speed. Explain why?

Solution: At a particular speed, the frequency of the  jerks to the various parts of vehicle becomes equal to their natural frequency. So resonance takes place. Hence, a loud rattling sound is produced.

4 Question: How are sound waves classified?

Solution: Sound waves are classified into three categories on the basis of frequency

• Audible sound (between20Hz-20,000 Hz)
• Ultrasound (above 20,000 Hz)
• Infra sound (below 20 Hz).

5 Question: Which method is used to produce high frequency ultrasonic waves? Discuss in details.

Solution: Piezoelectric Generator or Oscillator is used to produce high frequency of range 500MHz. Let us discuss the production of ultrasonic waves by this method.

Principle:

This is based on the Inverse piezoelectric effect. When a quartz crystal is placed under the effect of an alternating potential difference, vibrations produce in the crystal. If the frequency of electric oscillations coincides with the natural frequency of vibrations of the crystal, the vibrations will be of large amplitude. When the frequency of the electric field matches the ultrasonic frequency range, then the crystal produces ultrasonic waves.

Electric circuit :

Construction:

      It is common base NPN oscillator circuit.

      Coils L1  and  L2 are primary coils of the transformer.

      Coil L2  is connected to the collector and L1 connected to the base.

      The coil L1 and variable capacitor C1 form the tank circuit of the oscillator.

      Quartz crystal is placed between the metal plates A and B so as to form a parallel plate capacitor.

      Quartz Crystal is connected to the secondary coil L3 of the transformer through which output or ultrasonic wave is obtained.

      The frequency of the oscillations can be changed by changing the value of capacitance.

Working:

As soon as the circuit is closed, the current starts flow through the circuit and charges the capacitor. After the charging is completed the capacitor starts discharging through the inductor L1, this energy is stored in the form of electric field in capacitor C1  and in the form of magnetic field in inductor L1.

Due to this high frequency electric oscillations are produced in the tank circuit. Transistor is also produces electric oscillations. This energy or oscillations is provided to the secondary coil which is again fed to the quartz crystal. Thus oscillating electric field is converted to mechanical vibration of crystal.             

When the frequency of electric oscillations is equal to that of natural frequency of the crystal, resonance is achieved and the sound waves of maximum amplitude are produced. Thus by using inverse piezoelectric effect high frequency ultrasonic waves are produced.

Condition for Resonance:

Frequency of the oscillator circuit = Frequency of the vibrating crystal

Where, 

L1 is inductance of the circuit

C1 is capacitance of the circuit

t = Thickness of crystal slab

Y = Young's Modulus of material

ρ = Density of material

k = 1, 2, 3 ... (Integer Multiple)

Merits:

• High frequency Ultrasonic waves can be produced.
• This method is more effective than the Magnetostriction oscillator.
• The output power is very high
• We are able to get a stable and constant frequency of ultrasonic waves.
• It is not affected by temperature humidity

Demerits:

• Quartz crystal is very costly.
• Cutting and shaping the crystal is difficult.

6 Question: What is meant by ultrasonic?

Solution: Ultrasonics are the sound waves of frequency above audible range (i.e) above 20,000 Hz (or) 20 KHZ. This sound wave cannot be heard by human ear, but it has many useful applications in engineering and medical fields.

7 Question: Are the ultrasonic waves electromagnetic waves? Give proper reasons.

Solution: Ultrasonic waves are not electromagnetic waves because they are sound wave which does not consist of electric and magnetic vectors as in electromagnetic waves.

8 Question: Why are ultrasonic waves not audible to humans?

Solution: The audible range of frequencies for human beings is between 20HZ to 20,000HZ. Since the frequency of ultrasonic wave is having above 20,000HZ, it is not audible to humans.

9 Question: Why not ultrasonics be produced by passing high frequency alternating current through a loud speaker?

Solution: Ultrasonics cannot be produced by passing high frequency alternating current through loud speaker due to the following reasons.

• Loud speaker cannot vibrate at such high frequency.
• Inductance of the speaker coil becomes so high and practically no current flows through it.

10 Question: Mention the properties of ultrasonic waves?

Solution: Properties

a)     The ultrasonic waves cannot travel through vacuum.

b)    These waves travel with speed same as sound wave travel in any given medium.

c)     In homogeneous medium the velocity of ultrasonic wave is constant.

d)    These waves can also weld some material like plastics and metals.

e)     They have high energy content.

f)       Ultrasonic waves get reflected, refracted and absorbed just like sound waves.

g)    They can be transmitted over large distances without any appreciable loss of energy.

h)     They produce intense heating effect when passed through a substance.

i)       The ultrasonic waves have high frequency.

j)       Because of their smaller wavelength Ultrasonic waves produce negligible diffraction effects.

k)     Ultrasonic waves can produce vibrations in low viscosity liquids.

l)       When the ultrasonic wave is absorbed by a medium, it produces heat because of high frequency and high energy and that energy is used to drill and cut thin metals.

11 Question: Can we use a copper rod in a Magnetostriction generator? Why?

Solution: No, copper rod cannot be used to produce ultrasonics in magnetostriction generator because it is not a ferromagnetic material.

12 Question: Discuss Magnetostriction generator to produce ultrasonic wave? Also discuss its merits and demerits?

Solution: Magneto-striction generator or oscillator

Principle: Magnetostriction effect: Magnetostriction is a property of magnetic materials like nickel or iron that causes them to change their shape or dimensions during the process of magnetization. i.e. when these material is placed in magnetic field parallel to its length it undergoes  changes in its dimensions.  This is called Magnetostriction effect.

Electronic circuit:

Construction

      In above figure we are using NPN Transistor

      In which battery is connected in such a way that emitter is forward biased and collector is reverse biased.

      Current can be produced by applying necessary biasing to the transistor with the help of the battery.

      The current produced in a circuit can be noted by the mill ammeter connected across the coil L.

      The ends of the ferromagnetic rod A and B is wound by the coils L1 and L.

      The coil L1 is connected to the base of NPN transistor The coil L is connected to the collector of the NPN transistor as shown in the figure.

      The frequency of the oscillatory circuit (LC) can be adjusted by the condenser C.
 

Working 

The rod is initially magnetized by the DC power supply. The transistor is properly biased. The battery is switched on and hence current is produced by the transistor. This current is passed through the coil L, this current causes a change in the magnetization of the rod. Now, the rod starts vibrating due to Magnetostriction effect.

When rod is vibrating and the coil is wounded over a vibrating rod, An emf is induced in coil L, this induces an emf to coil L1 & a part of it is feed as input to the base. Hence, this feedback system makes transistor operates continuously. The e.m.f. Induced in the coil called as converse Magnetostriction effect. In this way the current is maintained in transistor so as the vibrations.

The frequency of the oscillatory circuit is adjusted by the condenser C and when this frequency is equal to the frequency of the vibrating rod, resonance occurs. At resonance, the rod vibrates longitudinally with larger amplitude producing ultrasonic waves of high frequency along both ends of the rod. 

Condition for resonance

Frequency of the oscillatory circuit = Frequency of the vibrating rod 

Where, 

L is inductance of the circuit

C is capacitance of the circuit

l is the length of the rod.
E is the young’s modulus of the material of the rod.
ρ is the density of material of the rod.

Merits:

• This Oscillatory circuit is simple to construct.
• Magnetostrictive materials are easily available at low cost
• Large output power can be generated by using this method. 

Demerits:
 

• It can produce frequencies up to 3 MHz only.
• As rod depends on temperature and the degree of magnetization so it becomes difficult to get a constant single frequency.
• As the frequency is inversely proportional to the length of the vibrating rod, so if you increase the frequency, the length of the rod gets decreased which is practically impossible.

13 Question: Define dielectric constant?

Solution: Dielectric constant it is a quantity measuring the ability of a substance to store electrical energy in an electric field.

Or

The ratio of the permittivity of the substance to the permittivity of the free space

Dielectric constant is also called Relative Permittivity Dielectric Constant is expressed by Greek letter kappa ‘κ’. It is dimensionless quantity.

It is mathematically expressed as:

κ =

Where,

• κ is the dielectric constant
• 𝜺 is the permittivity of the substance
• 𝜺0 is the permittivity of the free space.

14 Question: Define polarization of a dielectric material?

Solution: The diploe moment per unit volume of dielectric material is called the electric polarization of dielectric. It is represented by vector P.

In S.I. System, unit of polarization is C/m2.

15 Question: Define dielectric breakdown and dielectric strength.

Solution: Whenever the electrical field strength applied to a dielectric exceeds a critical value, very large current flows through it. The dielectric loses its insulating property and becomes conducting. This phenomenon is known as dielectric breakdown.

The electrical field strength at which dielectric breakdown occurs is known as dielectric Strength.

16 Question: What are the differences between polar and non-polar molecules? Also discuss the effect of electric field on dielectric?

Solution: Polar Molecules: Polar Molecules are those in which centre of gravity of positive and negative charge does not coincide with each other. This is because they all are asymmetric in shape. Examples: H2O, CO2, NO2 etc.
 

Non-Polar Molecule: Non-polar molecules are those in which centre of gravity of positive charge and negative charge coincide with each other. The molecule has zero dipole moment as they all are symmetric in shape. Examples: O2, N2, H2 etc.

Response of Dielectric to External Electric Field

In case of Polar Molecules -When the electric field is not present, it causes the electric dipole moment of these molecules in a random direction. This is why the average dipole moment is zero. If the external electric field is present, the molecules align themselves in the direction electric field and resulted in having dipole moment.

In case of Non Polar Molecules – As we know nonpolar molecule has zero dipole moment. In spite of zero dipole, when a dielectric nonpolar material is placed in an electric field. The positive and the negative charges in a nonpolar molecule experience forces in opposite directions. This force causes the separation between the charges and hence nonpolar molecule experiences induced dipole moment.

Figure a (left) Non Polar Molecule    Figure b (right) Polar Molecule

17 Question: What is meant by piezo-electric effect?

Solution: In Piezoelectric “Piezo” derived from the Greek word “piezein” and the meaning of piezein is squeeze or press. So its name itself explaining the effect.

The piezoelectric effect transforms kinetic or mechanical energy into electrical energy, due to crystal deformation.  When material is compressed or squeezed, this Mechanical stress applied to material generates electricity.

Microphones, speakers, buzzers, pressure sensors, hydrophones and many other sensing types of devices use direct piezoelectric effect.

18 Question: What is meant by inverse piezo-electric effect?

Solution: Inverse piezoelectric effect – when electric field is applied to a crystal it align all dipoles present in the material in order. This realignment results in deformation of crystal i.e. converting electrical energy into kinetic or mechanical energy. Reversal of the piezoelectric effect is called the inverse piezoelectric effect.

Using the inverse piezoelectric effect we can develop devices that produce acoustic sound waves. Even medical ultrasound and sonar transducers use inverse piezoelectric effect.

19 Question: The forbidden energy gap of dielectrics is __________
a) Less than 1.2 eV
b) Greater than or equal to 1.2 eV
c) Less than 3 eV
d) Greater than or equal to 3 eV
Solution:  d
Dielectrics are the materials devoid or lacking of free charges. An ideal dielectric is one which is completely devoid of charges. They have a forbidden energy gap of >=3eV.

20 Question:  Dipoles are created when dielectric is placed in __________
a) Magnetic Field
b) Electric field
c) Vacuum
d) Inert Environment
Solution: b
When the dielectric is placed in an electric field, like between the parallel plates of a capacitor, the dipoles are created and they tend to align themselves parallel to the direction of the electric field.

21 Question:  What is the main difference in the quality of ultrasonic waves produced by piezo electric and magnetostriction method?

Solution:

22 Question:  What is meant by SONAR? Mention two applications of it?

Solution: SONAR is a device which stands for Sound Navigation and Ranging.

The principle of SONAR is based on the echo sounding technique of ultrasonics. It is the acoustical technique used for locating the objects like submarines or icebergs in sea, by transmitting high frequency sound pulse and receiving it after reflection from that object.

Applications

• To find the depth of sea.
• To detect the submarines.

23 Question: Show that Electronic polarization depends upon radius of atom?

Or

Show that Electronic polarization depends upon volume of atom?

Solution: Electronic polarization refers to the separation of centre of positive  charge and centre of negative charge in a material.This separation is caused by high electric field.

Figure (a) shows the​charge distribution of an atom in absence of electric field while figure (b) show the charge distribution in presence of external electrical field.

Let us consider a single atom of atomic number Z. +e coulomb is the charge of each proton in the nucleus and -e coulomb is the charge of each electron surrounds the nucleus. All electrons in the atom form a spherical cloud of negative charge surrounds the positively charged nucleus. The charge of nucleus is +Ze coulombs and charge of the negative cloud of electrons is -Ze coulombs.

Let us also assume that the negative charge of the electrons cloud is homogeneously distributed on a sphere of radius R. In the absence of external electric field, the center of this sphere and center of nucleus of the atom coincide.

When an external electric field E is applied to the atom. Because of this external electric field the nucleus of the atom is shifted towards negative intensity of the field and the electron cloud is shifted towards the positive intensity of the field.

As due to influence of external electric field the center of nucleus and center of electrons cloud are separated, there will be an attractive force between them according to Coulomb’s law.

Let us suppose x is distance of separation between positive charge nucleus and electron cloud.

Also Nucleus is considered as point charge. Hence, the electrostatic force acting on the nucleus =+EZe                                           …….(1)

As we know nucleus has been shifted from the center of electrons cloud by a distance x.

By using Gauss’s theorem

The force is only due to electron cloud acting upon nucleus would only be due to the portion of the cloud enclosed by the sphere of radius x. Portion outside the sphere of radius x does not apply any force on the nucleus.

Volume of the sphere of radius x = (4/3)πx3  and

Volume of the sphere of radius R = (4/3)πR3

Now total negative charge of the electron cloud is -Ze  …..(2)

Hence, the quantity of negative charge enclosed by the sphere of radius x is,

[-Ze/(4/3)πR3] * (4/3)πx3  = -Ze (x3/ R3)   ……..(3)

According to coulomb’s law = q1q2 /4 πR2
Here it becomes charge on electrons q1=-Ze (x3/ R3)

Charge on nucleus q2= Ze

So coulomb’s force = {-Ze (x3/ R3) * Ze}/4πƐox2 = Z2e2x/4πƐo R3  ……(4)

Note- magnitude is taken into account. Neglect negative sign.

i.e. At equilibrium Electrostatic force = Coulomb force .

EZe = Z2e2x/4πƐoR3   ……..(5)

Upon simplify

x = {4πƐoR3/Ze} E

Now dipole moment = either charge * separation between charges i.e. x

= Ze *{4πƐoR3/Ze} E

= 4πƐoR3E

Polarization is number of dipole moment per unit volume. Let us suppose N is the number of dipoles per unit volume so

Pe=4πƐoR3EN

So it is clear that polarization depends upon radius of atom or volume of atom and number of atoms present per unit volume.