Linear Equation in two variables

Linear Equation in two variables

A concise details accordingly class 9.

1. An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables.

2. A linear equation in two variables has infinitely many solutions.

3. The graph of every linear equation in two variables is a straight line.

4. x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.

5. The graph of x = a is a straight line parallel to the y-axis.

6. The graph of y = a is a straight line parallel to the x-axis.

7. An equation of the type y = mx represents a line passing through the origin.

8. Every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph of the linear equation.

Application of Derivatives

Exercise 6.1 Class 12

Application of Derivatives

Mathematics

Class 12

Exercise 6.1

Solution of

QNo. 3, 4 and 5

Work And Energy

Work And Energy

This video lecture consists of 

1. Definition of Work

2. Dimension of work

3. Unit of work

4. Definition of energy

5. Types of mechanical energy

6. Conservation of energy 

Electromagnetic Waves

Electromagnetic Spectrum

At the time Maxwell predicted the existence of electromagnetic waves, the only familiar electromagnetic waves were the visible light waves. The existence of ultraviolet and infrared waves was barely established. By the end of the nineteenth century, X-rays and gamma rays had also been discovered. We now know that, electromagnetic waves include visible light waves, X-rays, gamma rays, radio waves, microwaves, ultraviolet and infrared waves. The classification of em waves according to frequency is the electromagnetic spectrum. There is no sharp division between one kind of wave and the next. The classification is based roughly on how the waves are produced and/or detected.

(Figure The electromagnetic spectrum, with common names for various part of it. The various regions do not have sharply defined boundaries.)

We briefly describe these different types of electromagnetic waves, in order of decreasing wavelengths.

Radio waves

Radio waves are produced by the accelerated motion of charges in conducting wires. They are used in radio and television communication systems. They are generally in the frequency range from 500 kHz to about 1000 MHz. The AM (amplitude modulated) band is from 530 kHz to 1710 kHz. Higher frequencies upto 54 MHz are used for short wave bands. TV waves range from 54 MHz to 890 MHz. The FM (frequency modulated) radio band extends from 88 MHz to 108 MHz. Cellular phones use radio waves to transmit voice communication in the ultrahigh frequency (UHF) band. How these waves are transmitted and received is described in Chapter 15.

Microwaves

Microwaves (short-wavelength radio waves), with frequencies in the gigahertz (GHz) range, are produced by special vacuum tubes (called klystrons, magnetrons and Gunn diodes). Due to their short wavelengths, they are suitable for the radar systems used in aircraft navigation. Radar also provides the basis for the speed guns used to time fast balls, tennis- serves, and automobiles. Microwave ovens are an interesting domestic application of these waves. In such ovens, the frequency of the microwaves is selected to match the resonant frequency of water molecules so that energy from the waves is transferred efficiently to the kinetic energy of the molecules. This raises the temperature of any food containing water.

Microwave oven

The spectrum of electromagnetic radiation contains a part known as microwaves. These waves have frequency and energy smaller than visible light and wavelength larger than it. What is the principle of a microwave oven and how does it work?Our objective is to cook food or warm it up. All food items such as fruit, vegetables, meat, cereals, etc., contain water as a constituent. Now, what does it mean when we say that a certain object has become warmer? When the temperature of a body rises, the energy of the random motion of atoms and molecules increases and the molecules travel or vibrate or rotate with higher energies. The frequency of rotation of water molecules is about

2.45 gigahertz (GHz). If water receives microwaves of this frequency, its molecules absorb this radiation, which is equivalent to heating up water. These molecules share this energy with neighbouring food molecules, heating up the food.

One should use porcelain vessels and not metal containers in a microwave oven because of the danger of getting a shock from accumulated electric charges. Metals may also melt from heating. The porcelain container remains unaffected and cool, because its large molecules vibrate and rotate with much smaller frequencies, and thus cannot absorb microwaves. Hence, they do not get heated up.

Thus, the basic principle of a microwave oven is to generate microwave radiation of appropriate frequency in the working space of the oven where we keep food. This way energy is not wasted in heating up the vessel. In the conventional heating method, the vessel on the burner gets heated first, and then the food inside gets heated because of transfer of energy from the vessel. In the microwave oven, on the other hand, energy is directly delivered to water molecules which is shared by the entire food.

Infrared waves

Infrared waves are produced by hot bodies and molecules. This band lies adjacent to the low-frequency or long-wave length end of the visible spectrum. Infrared waves are sometimes referred to as heat waves. This is because water molecules present in most materials readily absorb infrared waves (many other molecules, for example, CO2, NH3, also absorb infrared waves). After absorption, their thermal motion increases, that is, they heat up and heat their surroundings. Infrared lamps are used in physical therapy. Infrared radiation also plays an important role in maintaining the earth’s warmth or average temperature through the greenhouse effect. Incoming visible light (which passes relatively easily through the atmosphere) is absorbed by the earth’s surface and re-radiated as infrared (longer wavelength) radiations. This radiation is trapped by greenhouse gases such as carbon dioxide and water vapour. Infrared detectors are used in Earth satellites, both for military purposes and to observe growth of crops. Electronic devices (for example semiconductor light emitting diodes) also emit infrared and are widely used in the remote switches of household electronic systems such as TV sets, video recorders and hi-fi system.

Visible rays

It is the most familiar form of electromagnetic waves. It is the part of the spectrum that is detected by the human eye. It runs from about 4 × 1014 Hz to about 7 × 1014 Hz or a wavelength range of about 700 – 400 nm. Visible light emitted or reflected from objects around us provides us information about the world. Our eyes are sensitive to this range of wavelengths. Different animals are sensitive to different range of wavelengths. For example, snakes can detect infrared waves, and the ‘visible’ range of many insects extends well into the utraviolet.

Ultraviolet rays

It covers wavelengths ranging from about 4 × 10‐⁷ m (400 nm) down to 6 × 10‐¹⁰m (0.6 nm). Ultraviolet (uv) radiation is produced by special lamps and very hot bodies. The sun is an important source of ultraviolet light. But fortunately, most of it is absorbed in the ozone layer in the atmosphere at an altitude of about 40 – 50 km. uv light in large quantities has harmful effects on humans. Exposure to UV radiation induces the production of more melanin, causing tanning of the skin. UV radiation is absorbed by ordinary glass. Hence, one cannot get tanned or sunburn through glass windows.

Welders wear special glass goggles or face masks with glass windows to protect their eyes from large amount of UV produced by welding arcs. Due to its shorter wavelengths, UV radiations can be focussed into very narrow beams for high precision applications such as LASIK (Laser-assisted in situ keratomileusis) eye surgery. UV lamps are used to kill germs in water purifiers. 

Ozone layer in the atmosphere plays a protective role, and hence its depletion by chlorofluorocarbons (CFCs) gas (such as freon) is a matter of international concern. 

X-rays

Beyond the uv region of the electromagnetic spectrum lies the x-ray region. We are familiar with x-rays because of its medical applications. It covers wavelengths from about 10‐⁸ m (10 nm) down to 10‐¹⁰ m 

(10‐⁴ nm). One common way to generate X-rays is to bombard a metal target by high energy electrons. X-rays are used as a diagnostic tool in medicine and as a treatment for certain forms of cancer. Because x-rays damage or destroy living tissues and organisms, care must be taken to avoid unnecessary or over exposure. 

Gamma rays

They lie in the upper frequency range of the electromagnetic spectrum and have wavelengths of from about 10‐¹⁰m to less than 10‐¹⁰m. This high frequency radiation is produced in nuclear reactions and 

also emitted by radioactive nuclei. They are used in medicine to destroy cancer cells. 

Table summarises different types of electromagnetic waves, their production and detections. As mentioned earlier, the demarcation between different regions is not sharp and there are overlaps.

Some Basic Concepts of Chemistry

1. Some Basic Concepts of Chemistry
   

The study of chemistry is very important as its domain encompasses every sphere of life. Chemists study the properties and structure of substances and the changes undergone by them. All substances contain matter, which can exist in three states – solid, liquid or gas. The constituent particles are held in different ways in these states of matter and they exhibit their characteristic properties. 

Classification of Matter

Matter can also be classified into elements, compounds or mixtures. An element contains particles of only one type, which may be atoms or molecules. The compounds are formed where atoms of two or more elements combine in a fixed ratio to each other. Mixtures occur widely and many of the substances present around us are mixtures.

When the properties of a substance are studied, measurement is inherent. 

Measurements of Substances in SI System

The quantification of properties requires a system of measurement and units in which the quantities are to be expressed. Many systems of measurement exist, of which the English and the Metric Systems are widely used. The scientific community, however, has agreed to have a uniform and common system throughout the world, which is abbreviated as SI units (International System of Units).

Since measurements involve recording of data, which are always associated with a certain amount of uncertainty, the proper handling of data obtained by measuring the quantities is very important. The measurements of quantities in chemistry are spread over a wide range of 10-31 to 1023. Hence, a convenient system of expressing the numbers in scientific notation is used. The uncertainty is taken care of by specifying the number of significant figures, in which the observations are reported. The dimensional analysis helps to express the measured quantities in different systems of units. Hence, it is possible to interconvert the results from one system of units to another.

Basic Laws of Chemical Combination

The combination of different atoms is governed by basic laws of chemical combination — these being the 

1. Law of Conservation of Mass, 2. Law of Definite Proportions, 3. Law of Multiple Proportions, 

4. Gay Lussac’s Law of Gaseous Volumes and 5. Avogadro Law. 

All these laws led to the Dalton’s atomic theory, which states that atoms are building blocks of matter. 

Atomic and Molecular Mass

The atomic mass of an element is expressed relative to 12C isotope of carbon, which has an exact value of 12u. Usually, the atomic mass used for an element is the average atomic mass obtained by taking into account the natural abundance of different isotopes of that element. The molecular mass of a molecule is obtained by taking sum of the atomic masses of different atoms present in a molecule. The molecular formula can be calculated by determining the mass per cent of different elements present in a compound and its molecular mass.

Avogadro Number

 

The number of atoms, molecules or any other particles present in a given system are expressed in the terms of Avogadro constant (6.022 × 1023). This is known as 1 mol of the respective particles or entities.

Stoichiometric Calculations

Chemical reactions represent the chemical changes undergone by different elements and compounds. A balanced chemical equation provides a lot of information. The coefficients indicate the molar ratios and the respective number of particles taking part in a particular reaction. The quantitative study of the reactants required or the products formed is called stoichiometry. Using stoichiometric calculations, the amount of one or more reactant(s) required to produce a particular amount of product can be determined . 

The Amount of Substance in a Solution

The amount of substance present in a given volume of a solution is expressed in number of ways, e.g., 1. Mass Per cent, 2. Mole fraction, 3. Molarity and 4. Molality.

Electric Motor

Electric Motor

An electric motor is a rotating device that converts electrical energy into mechanical energy.

 

Uses of an Electric Motor

Electric motor is used as an important component in electric fans, refrigerators, mixers, washing machines, computers, MP3 players etc. 

Construction of an Electric Motor

An electric motor consists of a rectangular coil ABCD of insulated copper wire. The coil is placed between the two poles of a magnetic field such that the arm AB and CD are perpendicular to the direction of the magnetic field. The ends of the coil are connected to the two halves P and Q of a split ring. The inner sides of these halves are insulated and attached to an axle. The external conducting edges of P and Q touch two conducting stationary brushes X and Y, respectively, as shown in the Fig.

Working of an Electric Motor

Current in the coil ABCD enters from the source battery through conducting brush X and flows back to the battery through brush Y. Notice that the current in arm AB of the coil flows from A to B. In arm CD it flows from C to D, that is, opposite to the direction of current through arm AB. On applying Fleming’s left hand rule for the direction of force on a current-carrying conductor in a magnetic field. We find that the force acting on arm AB pushes it downwards while the force acting on arm CD pushes it upwards. Thus the coil and the axle O, mounted free to turn about an axis, rotate anti-clockwise. At half rotation, Q makes contact with the brush X and P with brush Y. Therefore the current in the coil gets reversed and flows along the path DCBA. A device that reverses the direction of flow of current through a circuit is called a commutator. In electric motors, the split ring acts as a commutator. The reversal of current also reverses the direction of force acting on the two arms AB and CD. Thus the arm AB of the coil that was earlier pushed down is now pushed up and the arm CD previously pushed up is now pushed down. Therefore the coil and the axle rotate half a turn more in the same direction. The reversing of the current is repeated at each half rotation, giving rise to a continuous rotation of the coil and to the axle.

The commercial electric motors use 

(i) An electromagnet in place of permanent magnet; 

(ii) Large number of turns of the conducting wire in the current- carrying coil; and 

(iii) A soft iron core on which the coil is wound. The soft iron core, on which the coil is wound, plus the coils, is called an armature. This enhances the power of the motor.

Class 10 Physics Chapter wise Synopsis

Light: Reflection and Refraction

• Light seems to travel in straight lines.

 

• Mirrors and lenses form images of objects. Images can be either real or virtual, depending on the position of the object.

 

• The reflecting surfaces, of all types, obey the laws of reflection. The refracting surfaces obey the laws of refraction.

 

• New Cartesian Sign Conventions are followed for spherical mirrors and lenses.

 

• Mirror formula, 

1/v + 1/u = 1/f

, gives the relationship between the object-distance (u), image-distance (v), and focal length (f) of a spherical mirror.

 

• The focal length of a spherical mirror is equal to half its radius of curvature.

 

• The magnification produced by a spherical mirror is the ratio of the height of the image to the height of the object.

 

• A light ray travelling obliquely from a denser medium to a rarer medium bends away from the normal. A light ray bends towards the normal when it travels obliquely from a rarer to a denser medium.

 

• Light travels in vacuum with an enormous speed of 3×10⁸ m s⁻¹. The speed of light is different in different media.

 

• The refractive index of a transparent medium is the ratio of the speed of light in vacuum to that in the medium.

 

• In case of a rectangular glass slab, the refraction takes place at both air-glass interface and glass-air interface. The emergent ray is parallel to the direction of incident ray.

 

• Lens formula, 

1/v - 1/u = 1/f

, gives the relationship between the object-distance (u), image-distance (v), and the focal length (f) of a spherical lens.

 

• Power of a lens is the reciprocal of its focal length. The SI unit of power of a lens is dioptre.

The Human Eye and the Colourful World

• The ability of the eye to focus on both near and distant objects, by adjusting its focal length, is called the accommodation of the eye.

 

• The smallest distance, at which the eye can see objects clearly without strain, is called the near point of the eye or the least distance of distinct vision. For a young adult with normal vision, it is about 25 cm.

 

• The common refractive defects of vision include myopia, hypermetropia and presbyopia. Myopia (short-sightedness – the image of distant objects is focussed before the retina) is corrected by using a concave lens of suitable power. Hypermetropia (far-sightedness – the image of nearby objects is focussed beyond the retina) is corrected by using a convex lens of suitable power. The eye loses its power of accommodation at old age.

 

• The splitting of white light into its component colours is called dispersion.

 

• Scattering of light causes the blue colour of sky and the reddening of the Sun at sunrise and sunset.

Electricity

• A stream of electrons moving through a conductor constitutes an electric current. Conventionally, the direction of current is taken opposite to the direction of flow of electrons.

 

• The SI unit of electric current is ampere.

 

• To set the electrons in motion in an electric circuit, we use a cell or a battery. A cell generates a potential difference across its terminals. It is measured in volts (V).

 

• Resistance is a property that resists the flow of electrons in a conductor. It controls the magnitude of the current. The SI unit of resistance is ohm (Ω ).

 

• Ohm’s law: The potential difference across the ends of a resistor is directly proportional to the current through it, provided its temperature remains the same.

 

• The resistance of a conductor depends directly on its length, inversely on its area of cross-section, and also on the material of the conductor.

 

• The equivalent resistance of several resistors in series is equal to the sum of their individual resistances.

 R = R₁ + R₂ + R₃

• A set of resistors connected in parallel has an equivalent resistance R given by

1/R = 1/R₁ + 1/R₂ + 1/R₃

• The electrical energy dissipated in a resistor is given by

 

• W = V × I × t

 

• The unit of power is watt (W). One watt of power is consumed when 1 A of current flows at a potential difference of 1 V. 

• The commercial unit of electrical energy is kilowatt hour (kWh).

 1 kW h = 3,600,000 J = 3.6 × 10⁶ J.

Magnetic Effects of Electric Current

• A compass needle is a small magnet. Its one end, which points towards north, is called a north pole, and the other end, which points towards south, is called a south pole.

 

• A magnetic field exists in the region surrounding a magnet, in which the force of the magnet can be detected.

 

• Field lines are used to represent a magnetic field. A field line is the path along which a hypothetical free north pole would tend to move. The direction of the magnetic field at a point is given by the direction that a north pole placed at that point would take. Field lines are shown closer together where the magnetic field is greater.

 

• A metallic wire carrying an electric current has associated with it a magnetic field. The field lines about the wire consist of a series of concentric circles whose direction is given by the right-hand rule.

 

• The pattern of the magnetic field around a conductor due to an electric current flowing through it depends on the shape of the conductor. The magnetic field of a solenoid carrying a current is similar to that of a bar magnet.

 

• An electromagnet consists of a core of soft iron wrapped around with a coil of insulated copper wire.

 

• A current-carrying conductor when placed in a magnetic field experiences a force. If the direction of the field and that of the current are mutually perpendicular to each other, then the force acting on the conductor will be perpendicular to both and will be given by Fleming’s left-hand rule. This is the basis of an electric motor. An electric motor is a device that converts electric energy into mechanical energy.

 

• The phenomenon of electromagnetic induction is the production of induced current in a coil placed in a region where the magnetic field changes with time. The magnetic field may change due to a relative motion between the coil and a magnet placed near to the coil. If the coil is placed near to a current-carrying conductor, the magnetic field may change either due to a change in the current through the conductor or due to the relative motion between the coil and conductor. The direction of the induced current is given by the Fleming’s right-hand rule.

 

• A generator converts mechanical energy into electrical energy. It works on the basis of electromagnetic induction.

 

• In our houses we receive AC electric power of 220 V with a frequency of 50 Hz. One of the wires in this supply is with red insulation, called live wire. The other one is of black insulation, which is a neutral wire. The potential difference between the two is 220 V. The third is the earth wire that has green insulation and this is connected to a metallic body deep inside earth. It is used as a safety measure to ensure that any leakage of current to a metallic body does not give any severe shock to a user.

 

• Fuse is the most important safety device, used for protecting the circuits due to short-circuiting or overloading of the circuits.

Thanks to Watch and stay with me till last words of synopsis.