Class Notes Nepal 2078

Electromagnetic Induction| Electricity and Magnetism | Chapter 5 | Class 12 | New Curriculam 2078 | Notes | Nepal |

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Magnetic Field | Electricity and Magnetism | Chapter 3 | Class 12 | New Curriculam 2078 | Notes | Nepal |

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MODEL QUESTION PAPER 2078 / 2079 || COMPUTER || NEB CLASS 12 || NEPAL

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Thermoelectric Effects | Electricity and Magnetism | Chapter 2 | Class 12 | New Curriculam 2078 | Notes | Nepal |

2021-11-27T16:26:00.028+05:45

Thermoelectric Effects

The phenomenon of production of electricity when two junction of different metal plates are kept at different temperature is known as thermoelectric effect.

1. Seebeck’s Effect

2. Peltier Effect

3. Thomson’s Effect

Seebeck Effect

The phenomenon of generation of an electric current in a thermo couple by keeping its two end at different temperature is known as seebeck effect.

Thermocouple:

A pair of wires of different metals forming a loop and producing thermo electricity is known as thermo couple. Iron-copper; Antimony-Bismuth are some example of thermocouple.

Thermoelectric series

An arrangement of metals in series in which any two metals can be used to form a thermocouple is known as thermo electric series. These series contains the metal Antimony, iron, zinc, silver, gold, tin, lead, copper, platinum, nickel, Bismuth.

i.e Sb, Fe, Zn, Ag, Au, Sn, Pb, Cu, Pt, Ni, Bi

Peltier Effect

When an electric current is passed through a thermocouple, heat is absorbed or released at the junction, depending upon the direction of current. This effect is called peltier effect.

Thomson Effect

The phenomenon of evolution or absorption of heat along the length of conductor on passing current through it, when its two ends are kept at different temperature is known as Thomson effect.

If electric current is passed through copper wire from hot to cold end, the heat is released and wire get heated then such conductor is known as positive Thomson effect. But in the case of iron, heat is absorbed and wire get cooled when current is passed from hot to cold end. So, iron is said to be negative thomson’s effect.

Variation of thermo emf with temperature

Let us consider a iron-copper thermo-couple to study the variation of thermo emf with temperature. One junction of the thermocouple is immersed into hot oil bath and another junction is kept into cold ice. When the temperature of both junction are at same temperature 0 C, there is no flow of current and no emf is produced. When the temperature of oil bath is increased keeping cold junction at 0 C, the emf is produced and becomes maximum at temperature `\theta_n` called neutral temperature.

When the temperature of oil bath is increased beyond neutral temperature, the thermo emf goes on decreasing and becomes zero at temperature `\theta_i` called the temperature of inversion. Finally the direction of thermo emf is changed into opposite direction. If `\theta_c` be the temperature of cold junction then,

`\theta_i - \theta_n = \theta_n -\theta_c`

`2\theta_n = \theta_c + \theta_i`

`\therefore \theta_n = \frac{\theta_c + \theta_i}{2}`

Neutral temperature is the mean of temperature of cold junction & temperature of inversion.

The variation of thermo emf(E) with temperature (`\theta`) is given by

`E = \alpha\theta + \frac{1}{2} \beta \theta^2`

Where `\alpha` and `\beta` are constant.

Relation between `\alpha` , `\beta` , `\theta_n` and `\theta_i` :

The equation of thermo emf with temperature is given by

`E = \alpha\theta + \frac{1}{2} \beta \theta^2`

`\frac{dE}{d\theta} = \frac{d(\alpha\theta+ \frac{1}{2}\beta\theta^2)}{d\theta}`

`\frac{dE}{d\theta} = \alpha + \beta\theta`

E is maximum at `\theta` = `\theta_n` and so `\frac{dE}{d \theta} =0`

`0 = \alpha + \beta\theta_n`

`\theta_n = -\frac{\alpha}{\beta}`

thermo emf(E) = 0 at `\theta = \theta_i` and

`E = \alpha\theta_i + \frac{1}{2} \beta (\theta_i)^2`

`\frac{1}{2}\beta(\theta_i)^2 = -\alpha\theta_i`

`\theta_i \neq 0`,

`frac{1}{2}\beta(\theta_i)^2 = - \alpha\theta_i`

`\theta_i = -\frac{2\alpha}{\beta}`

Neutral temperature (`\theta_n`) : The temperature of hot junction at which the thermo emf becomes maximum is known as neutral temperature. It depends upon the nature of metals. It is independent of temperature of junction.

Temperature of inversion (`\theta_i`): The temperature of hot junction at which the thermo emf becomes zero and reverses its direction is known as temperature of inversion. It depends upon the nature of metals and the temperature of cold junction.

Magnitude of thermo emf depends on the nature of the metals forming a thermocouple and difference in temperature of two junction.

Application of thermoelectric effect:

1. Thermopile:

Thermopile is a device used for the detection and measurement of heat radiation. It works on the principle of thermoelectric effect.

Electrical Circuits | Electricity and Magnetism | Chapter 1 | Class 12 | New Curriculam 2078 | Notes | Nepal |

2021-11-16T23:02:00.042+05:45

Galvanometer

A galvanometer is an instrument used to detect small(feeble) current in electric circuit. Galvanometer may be moving magnet galvanometer or moving coil galvanometer. In laboratory, a moving coil galvanometer is used.

When a current is passed through the coil, it experience a torque and deflects. The deflection of coil is directly proportional to the strength of current. A pointer attached to the coil shows a deflection under measuring skill. It is denoted by

Shunt

A low (small) resistance which is connected in parallel to a galvanometer is called shunt. It carries most of the current passing through the circuit and thus leaving a small amount of current to flow through galvanometer.

Uses:

1. It is used to for converting galvanometer into ammeter

2. It is used to prevent damage of galvanometer

3. It reduces decreases the total resistance of the circuit.

Conversion of Galvanometer into Ammeter

An ammeter is an instrument used to measure the current in electric circuit. It is connected in series. If large current is passed, the galvanometer may damaged. So when galvanometer is converted into ammeter, it measures current without any damage to it.

To convert galvanometer into ammeter of range 0 to IA, a small resistance S is connected in parallel to the galvanometer which is called shunt. Let G be the resistance of galvanometer and I be the maximum current measured by ammeter. Then `I_g` be the current flowing through galvanometer. Then (`I-I_q`) is the current passing through the shunts.

Since G & S are parallel combination,

P.d across S = P.d across G

This is the value of shunt. Knowing the value of `I_g`, I and G, the value of S can be calculated. By connecting shunt of this value, the given galvanometer can be converted into ammeter of the range 0 to IA.

The effective resistance `R_a` of the ammeter is given by

`\frac{1}{R_a} = \frac{1}{G} + \frac{1}{S}`

`\frac{1}{R_a} = \frac{S+G}{GS}`

`R_a = \frac{GS}{G+S}`

In practice, G is large as compared to S. Since S is low resistance, equivalent resistance `R_a` of ammeter is very low and when it is connected in series in circuit, it will not affect the current passing through circuit.

Conversion of Galvanometer into an Voltmeter

A voltmeter is a device used to measure P.d. between two points in a circuit. It is connected in parallel. Voltmeter should have very high resistance ideally infinite resistance and must not change the current.

To convert a galvanometer into voltmeter of range 0 to V volts, a high resistance R is connected in series to the galvanometer which is called multiplier. The combination of galvanometer and multiplier for the conversion of voltmeter is as shown in fig.

Let G be the resistance of the galvanometer and `I_g` be the current flowing through galvanometer which produces the maximum deflection in the galvanometer. Since high resistance R is connected in series with galvanometer, same current `I_g` flows through Resistance R. The voltage across the voltmeter is

`V = (I_g)(R+G)`

`V = (I_g)R + I_gG`

`I_gR = V – I_g G`

`R = \frac{V}{I_g} – G`

This is the value of resistance R. Knowing the value of V, `I_g` and G, the value of R can be calculated. When the resistance of this value is connected in series to the galvanometer, it will work as a voltmeter of range 0 to V volt.

Eletrical Circuits

Kirchoff’s Laws:

First Law:

It states that the algebraic sum of currents at a junction of circuit is zero.

Symbolically

`\sum I = 0`

Current flowing towards the junction as positive and current flowing out from the junction as negative. Then applying Kirchoff’s 1^st law at the junction A, we have

`I_1 + I_2 + I_3 - I_4 - I_5 - I_6 = 0`

`I_1 + I_2 + I_3 = I_4 + I_5 + I_6`

Hence, the sum of current flowing towards the junction is equal to sum of current flowing out from the junction.

Second Law

It states that in a closed loop, the algebraic sum of emfs is equal to the algebraic sum of product of current and resistance.

Symbolically,

`\sum E = \sum IR`

Explanation:

ABCA be a closed circuit whose sides AB, BC and CA carry currents `I_1, I_2` and `I_3` respectively. The resistances of the sides are `R_1, R_2` and `R_3` respectively. There is a cell of emf E in the circuit. In complete circuit ABCA, we can write

`\sum E = \sum IR`

`I_1R_1+I_2R_2+I_3R_3=E`

If no cell in the circuit, E=0. So, we have

`I_1R_1 + I_2R_2+I_3R_3=0`

Wheatstone bridge

Wheatstone bridge is an electrical circuit which is used to find the unknown resistance. It consists of four resistance P,Q,R & X in which P,Q,R are known resistance and X is unknown resistance, when the battery is connected across B&D then at balanced condition i.e. `I_g=0`

`\frac{P}{Q} =\frac{X}{R}`

This is the balanced condition for wheatstone bridge and known as principle of wheatstone bridge.

Let, initially the circuit is not in balanced condition and suppose `I_1 , I_2, I_3` and `I_4` be the current passing through P,X,Q and R respectively. Then applying Kirchhoff’s first law at the junction B is

`\sum I = 0`

`I_1 – I_g – I_3 =0`

`I_1=I_g +I_3`

And at junction D,

`I_2 + I_g – I_4 =0`

`I_2 + I_g = I_4`

Applying Kirchhoff’s second law (voltage law) in loop ABDA

`\sum E = \sum IR`

`0 = I_1 P + I_g G – I_2 X`

`I_1 P + I_g G = I_2 X`

Similarly in closed loop BCDA,

`0 = I_2 Q – I_g G – I_4 R`

`I_3 Q = I_g G + I_4 R`

At balanced condition, the value of P, Q & R are so adjusted that no current passes through the galvanometer i.e. `I_g= 0` then above eqn becomes

`I_1 = I_3` and `I_2 = I_4`

And

`I_1 P = I_2 X`

`I_3 Q = I_4 R`

Dividing `eq^n` 6 by 7,

`\frac{I_1 P}{I_3 Q} = \frac{I_2 X}{I_4 R}`

From equation 5 & 8

`\frac{I_1 P}{I_1 Q} = \frac{I_2 X}{I_2 R}`

`\therefore \frac{P}{Q} = \frac{X}{R}`

This is required balanced condition of wheatstone bridge.

Meterbridge

Meterbridge is an electric device which is used to determine the unknown resistance. It works on the principle of wheatstone bridge.

It consists of three metal strips of copper which are fixed on wooden board. A constantan wire of uniform cross section and 1m length is stretched tight between two terminal of meter scale as shown in figure

A resistance box R is connected in one gap and unknown resistance ‘X’ is connected in another gap. A galvanometer is connected between point D & jockey over the wire then resistance of AB = P and BC = Q

When the bridge is balanced then, wheatstone principle gives

`\therefore \frac{P}{Q} = \frac{X}{R}`

`\therefore (Resistance of AB)/(Resistance of BC) = \frac{X}{R}`

Let length of wire (AC) = 100cm and AB = l cm & BC = (100-l) cm. Since the wire has uniform cross-section and `\rho` is constant then its resistance is directly proportional to the length i.e. `P \alpha l` and `Q \alpha (100-l)`

So,

`\frac{l}{100-l} = \frac{X}{R}`

`X = (\frac{l}{100-l})R`

By knowing the value of l, 100-l and R, the unknown resistance (X) can be calculated.

Potentiometer

Potentiometer is an electric device which is used to measure the emf of cell and internal resistance, to compare the emf of two cell and p.d. between two points. It is an ideal instrument for measuring the p.d. between two points because it does not draw any current from circuit.

Potentiometer is a long wire AB and uniform cross sectional area stretched between copper strips fixed on wooden board. The wire used in potentiometer is manganin or constantan. The length of wire is 5-10m and each of the segment length is 1m. Driving cell is connected across point A and B. The balanced point on wire is obtained by sliding the jockey over a wire. At balanced point there is no deflection on galvanometer.

Principle: It is based on the principle that steady current is passed through a wire of uniform cross sectional area, the p.d. across any portion(segment) of wire is directly proportional to its length.

`V \alpha l`

Application:

1. To find the internal resistance of given test cell:

Experimental arrangement to find the internal resistance of given cell using potentiometer is as shown in fig.

At first the key `K_1` is closed (with key `K_2` open) and jockey is slided over the potentiometer wire till the galvanometer shows null deflection. Suppose galvanometer shows null deflection at point `e` such that `AC = l_1`. Since `l_1` is the balancing length for emf of test cell so.

Emf (E) = `V_AC = k l_1`

K is potential gradient.

Now a suitable value of resistance R is taken from the resistance box and `k_2` key is closed. Then above process is repeated till the galvanometer shows null deflection. Let galvanometer shows null deflection at point D such that AD = `l_2`. Since `l_2` is the balancing length for terminal p.d. of the test cell so terminal p.d.(V) =`V_AD` = `k l_2`

Dividing equation 1 by 2

`\frac{E}{V} = \frac{k l_1}{k l_2} = \frac{l_1}{l_2}`

Also expression for internal resistance of a source in terms of its emf and terminal p.d. is

`r = R(\frac{E}{V} -1) = R(\frac{l_1}{l_2} -1)`

`\therefore r = R(\frac{l_1}{l_2} -1)`

By knowing the value of R, `l_1` and `l_2`, the value of internal resistance of given cell can be determined.

2. Compare the Emf of given cells:

The experimental arrangement to compare the emf of given cells is as shown in fig.

At first the test cell of emf `E_1` is connected in circuit through the key `k_2` and jockey is slided over the potentiometer wire till the galvanometer shows the null deflection. Let the galvanometer shows deflection at point C such that AC = `l_1`. Then the emf of first cell is given by

`E_1 = V_AC = kl_1`

K is potential gradient

Now first cell is disconnected and the second cell of Emf `E_2` is connected in the circuit through the key `k_2` and then repeat above process till the galvanometer shows null deflection. Suppose galvanometer shows null deflection at point D such that AD = `l_2`. Then emf of second cell is given by

`E_2 = V_AD = kl_2`

Dividing equation 1 by 2

`\frac{E_1}{E_2} = \frac{k l_1}{k l_2}`

`\frac{E_1}{E_2} = \frac{l_1}{l_2}`

By knowing the value of `l_1` and `l_2`, the emfs of two cell can be compared.

Mechanical Waves | Wave & Optics | Part 2 | Class 12 | New Curriculam 2078 | Notes | Nepal |

2021-11-04T13:20:00.051+05:45

Mechanical Waves

The waves which needs material medium to propagate is called mechanical waves. Example: water waves, sound waves, etc.

Mechanical waves are of two types:

1. Longitudinal Waves

2. Transverse waves

(Explanation of types of mechanical waves is already in part 1: click here )

Speed of Sound Wave in any medium:

Experimentally it is found that speed of sound wave in any medium depends on two factors; elasticity of medium and density of medium.

Let 'V' be the speed of sound in any medium, 'E' be the elasticity and 'p' be the density of the medium. Then,

`V \alpha E^a`-------------(1)
`V \alpha \rho^b` ---------(2)

Combining equation (1) and (2) we get,

`V \alpha E^a\rho^b`

`V = KE^a\rho^b`-----------(3)

Writing the dimension of each terms in eqn (3) we get,

`[M^0L^1T^-1]=[M^1L^-1T^-2]^a [M^1L^-3T^0]^b`

`[M^0L^1T^-1]=[M^(a+b)L^(-a-3b)T^(-2a)]`

Equating the power of M, L and T we get,

`a+b=0`, `-a-3b=1` and `-2a=-1`

Solving these equation we get,

`a=\frac{1}{2} and b=\frac{-1}{2}`

Putting the value of a and b in equation (3) we get,

`V = K E^\frac{1}{2} \rho^\frac{-1}{2}`
`V = K \sqrt\frac{E}{\rho}`

Experimentally the value of K is found to be 1 so,

`V = \sqrt\frac{E}{\rho}` which is the required expression for the velocity of sound wave in any medium.

Velocity of sound in solid:

Elasticity(E)= Young's Modulus of Elasticity(Y)

`\therefore` velocity of sound in solid(`v_s`)= `\sqrt\frac{Y}{\rho}`

For steel, `Y = 2\times10^11 Nm^(-2)`

`\rho = 7800 kgm^(-3)`

`\therefore` velocity of sound in steel (`V_steel`)= `\sqrt\frac{2\times10^11}{7800}`

= 5064m/s

Velocity of sound in liquid

Elasticity(E) = Bulk modulus of Elasticity (`\beta`)

`\therefore` velocity of sound in liquid (`v_l`)= `\sqrt\frac{\beta}{\rho}`

For water, `\beta` = 2.04`\times10^9Nm^(-2)`

`\rho = 1000kgm^(-3)`

`\therefore` velocity of sound in water (`v_w`) = `\sqrt\frac{2.04\times10^9}{1000}`

= 1428 m/s

Velocity of sound in gaseous medium

i) Newton's formula for the velocity of sound in gas:

Elasticity(E) =Bulk modulus of elasticity(`\beta`) = Pressure exerted by gas (P)

`\therefore` velocity of sound in gas (`v_g`) = `\sqrt\frac{P}{\rho}`

At NTP, P = 1.013 `\times 10^5` N/`m^-2` and `\rho` = 1.29 kg/`m^3`

`\therefore` velocity of sound in gas (`v_g`) =`\sqrt\frac{1.013\times10^5}{1.29}`

= 280 m/s which does not match with the experimental result.

ii) Laplace correction for the velocity of sound in gas:

Elasticity(E)= Bulk modulus of Elasticity (`\beta`) = `\gammaP`

`\therefore` velocity of sound in gas (`v_g`) = `\sqrt\frac{\gammaP}{\rho}`

At NTP, P = 1.013 `\times 10^5` N/`m^-2` , `\rho` = 1.29 kg/`m^3` and `\gamma`=1.4

`\therefore` velocity of sound in gas (`v_g`) =`\sqrt\frac{1.4\times1.013\times10^5}{1.29}`

= 332 m/s which match with the experimental result.

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Chemical Analysis | Volumetric Analysis | Part 1 | Class 12 | New Curriculam 2078 | Notes | Nepal |

2021-11-02T19:56:00.006+05:45

Chemical Analysis

The analysis which is done for the detection of main constituents present in the given chemical compound.

Qualitative Analysis

The analysis which is done for the detection of ions or radicals present in the given inorganic salt is known as qualitative analysis.

Quantitative Analysis

The analysis which is done by measuring the quantity of a particular component present in the given inorganic salt is known as quantitative analysis.

Gravimetric analysis

The word 'gravimetric' derived from Latin word; 'gravis' means weighing and 'metric' means measurement.

Gravimetric analysis is the part/method of analytical analysis to determine the quantity of analyte based on the mass of solid.

Example: Measuring the solids suspended in the water sample: once a known volume of water is filtered, the collected solids are weighed.

Steps in gravimetric analysis

1. 1.   Preparation of a solution
2.       Precipitation
3.       Filtration
4.       Washing
5.       Drying and ignition
6.       Heating to constant weight

1. Preparation of solution: A definite quantity of a substance under consideration is taken and it's solution is prepared in a suitable solvent.

2. Precipitation: The constituent to be determined is precipitated by adding an appropriate reagent (precipitant) in slight excess to a dilute hot solution of the weighted sample with constant stirring. The insoluble precipitate should be highly insoluble and be easily filterable.

3. Filtration: It is the process of the separation of precipitate from the mother liquor. If can be done either by filter paper sintered glass crucible etc

4. Washing: Precipitate should be washed several times with a suitable wash liquid. Washing should be continued till the filtrate is free from specified ions.

5. Drying and ignition: It is done to remove solvent and wash electrolytes and is done by heating at 110°C to 120°C for 1 to 2 hours

Heating to constant weight: After the precipitate is allowed to cool (preferably in a desiccators to keep it from absorbing moisture).

Volumetric Analysis

It is a general term for a method in quantitative analysis in which the amount of substance is determined by the measurement of the volume that the substance occupies.

It is commonly used to determine the unknown concentration of known reactant.

It is ofter referred to as titration, a laboratory technique in which one substance of known concentration and volume is used to react with another substance of known concentration.

Titration

The process in which the concentration of unknown solution is determined with the help of standard solution by using the indicator is called titration.

Titrant

The solution of known concentration which is usually taken in a burette is called titrant.

Titrand

The solution of unknown concentration which is being titrated and usually taken in a conical flask is called titrand.

Titration is the slow addition of one solution of a known concentration (called a titrant) to a known value of another solution of unknown concentration until the reaction reaches neutralization, which is often indicated by a color change

Wave Motion | Wave & Optics | Part 1 | Class 12 | New Curriculam 2078 | Notes | Nepal |

2021-11-02T10:05:00.040+05:45

Waves

The disturbance which propagates from one point to another point without net transport of matter is called wave. It is due to repeated simple harmonic motion of particle carrying energy but not due to actual linear movement of particle carrying energy.

Examples: sound wave, light wave, water ripples etc.

Characteristics of Wave Motions:

Wave motion is a disturbance propagating in a medium.
It transfers energy as well as momentum from one point to another.
When it propagates in a medium particles of the medium execute vibrating
motion about their mean position.
It has finite and fixed speed which depends on the nature of the medium and
is given by v = 𝑓 × 𝜆.
5. Speed of wave and particle velocity are different. The particle velocity is 𝑣 =𝑎𝜔𝑐𝑜𝑠𝜔𝑡.

Types of wave motion

1. Electromagnetic waves

2.Mechanical waves

3.Matter waves

Electromagnetic waves

1. Doesn't need a medium for its propagation. light wave, heat waves, radio waves etc.

2. Speed is equal to speed of light.

3. Moving charges back and forth will produce oscillating electric and magnetic fields. The electric field and magnetic field are perpendicular to each other.

Mechanical waves:

• Requires material medium for their propagation.

• For the propagation of mechanical waves, medium must possess elasticity, inertia and low resistance for motion (i.e. damping must be very small)

• Examples: water waves, sound waves, waves in pipes and strings etc.

Types of mechanical waves

Transverse waves:

If the particle of a medium vibrate perpendicularly to the propagation of the wave, then the wave is called transverse wave. For eg: wave on strings, water ripples, etc.

• Vibration is perpendicular to the propagation of waves.

• For the propagation of transverse wave medium should havemodulus or rigidity. So, these waves are present in solids.

• Polarization is possible.

• Travels in the form of crest and troughs.

Longitudinal waves:

If the particles of a medium vibrate along the direction of propagation of the wave, the wave is called longitudinal wave. For eg: waves on spring along length, sound waves on air, etc.

• Vibration is parallel to the propagation of waves.

• For the propagation of longitudinal waves medium must have Bulk modulus of elasticity. So, these waves are present in all three media solid liquid and gases.

• Polarization is not possible.

• Travels in the form of compression and rarefactions.

Matter Wave

The waves associated with the microscopic particles such as electrons, protons, neutrons, atoms and molecules when they are in motion are called matter waves. The concept of matter wave was first introduced by de-Broglie, so it is called de-Broglie wave

Waves on the water:

Water waves are an example of waves that involve a combination of both longitudinal and transverse motions. As a wave travels through the water, the particles travel in clockwise circles. The radius of the circles decreases as the depth into the water increases.
Notes: Waves on the earth surface during earthquake is the combination of longitudinal and transverse waves.

Some terms related to wave:

Displacement:

The distance of an oscillating particle of the wave from its equilibrium position at any particular time.

The displacement equation is,

𝒚 = 𝐚𝐬𝐢𝐧 𝒘𝒕 − 𝜱

Where, y= displacement of the particle at any instant of time ‘t’

a= amplitude=maximum displacement from its mean position.

w= angular frequency=2𝜋𝑓

𝑤𝑡 − 𝛷=phase of the wave

Particle speed:

During wave propagation medium of the particles undergo simple harmonic motion

around their mean position. The speed of these particles is called particle speed.

We know,

𝒚 = 𝐚𝐬𝐢𝐧(𝒘𝒕 − 𝜱)

Differentiating with respect to time we get,

𝑎𝑤𝑐𝑜𝑠(𝒘𝒕 − 𝜱)

∴ 𝒗 = 𝒂𝒘𝒄𝒐𝒔 𝒘𝒕 − 𝜱

For maximum speed, 𝑐𝑜𝑠(𝒘𝒕 − 𝜱) =1 so = 𝑎𝑤 (𝑚𝑒𝑎𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛)

For minimum speed. 𝑐𝑜𝑠 (𝒘𝒕 − 𝜱) =0, so = 0 (𝑒𝑥𝑡𝑟𝑒𝑚𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛)

Frequency:

Number of complete oscillation made by oscillating particles in one second is called

frequency. It is denoted by ‘f’. Its SI unit is Hertz(per second).

Wavelength

The distance between successive identical parts of the wave. It's unit is meters. Also,

The linear distance travelled by wave in one complete oscillation is called wavelength.

Time period

The time taken for one complete oscillation is called time period.

It is denoted by T

Or,

It is also defined as the reciprocal of frequency.

i.e.

Its SI unit is seconds(s)

Wave Velocity

Linear distance travelled by wave per unit time is called wave velocity.

i.e. 𝑤𝑎𝑣𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑤𝑎𝑣𝑒 𝑙𝑒𝑛𝑔𝑡ℎ / 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑

∴ 𝑣 = 𝜆 × 𝑓

Its SI unit is

Phase of the wave:

The argument of the sine/cosine in a given wave equation is called phase angle or simply phase of the wave. It gives the state of the vibrating particle as regards its position and direction of motion.

The equation of progressive wave is,

Here, phase of the wave(𝜑) =

Its SI unit is radian.

Progressive Waves

The waves which moves in forward direction with constant amplitude is called progressive waves.

Characteristics of Progressive Waves:

A progressive waves transfer energy from one part of space at other
In progressive waves, no particle is permanently at rest.
In progressive waves all the particles vibrates with same amplitude and with same frequency.
It may be either transverse or longitudinal.

Equation of progressive waves and equation of stationery wave equation:

From Book

Principle of superposition of waves.

The principle of superposition of waves state, "if two or more that two waves superimposed, then the displacement equation of the resultant waves is equal to the algebraic sum of the displacement of the individual waves that superimposed.

Let be the displacement equation of the individual waves that superimposed then the displacement of the resultant wave is :

*Thank You*