Category: Physics Notes PPT

Ionizing radiation is a type of energy released by atoms that travel in the form of particles (alpha, beta, or neutrons) or electromagnetic waves (X-rays or gamma rays). The energy emitted is in the form of ionizing radiation. Radioactivity is the spontaneous emission of radiation in the form of high energy photons resulting from a nuclear reaction. It is a random process that occurs at the level of individual atoms. Radioactive substances like thorium, uranium, and radium produce radiation and they also produce a lot of energy. They all can easily knock electrons out of atoms and form charged particles.

Ionizing Radiation Definition

Ionizing radiation is radiation with great energy so that during an interaction with the atom, it can remove tightly bound electrons from the orbit of an atom, causing the atom to be changed from their neutral state. Ionizing radiation occurs in two forms- waves or particles. It is made up of ions, atoms, or energetic subatomic particles moving at high speeds and electromagnetic waves on the high-energy end of the electromagnetic spectrum.

X-rays, gamma rays and the ultraviolet part of the electromagnetic spectrum. It has more energy than non-ionizing radiation, enough to cause chemical changes by breaking bonds. 

There are 3 main types of ionizing radiation:

• Alpha particles 

• Beta particles

• Gamma rays 

Alpha Particles

Alpha particles are particulate radiations with hugely ionizing form. Alpha particles are slower and heavier than x-rays and gamma rays. These particles become dangerous when they are inhaled. Radon is odorless, colorless, and tasteless gas which comes from the decay of the element radium. The alpha particles from radon are about 20 times as effective as X rays and gamma rays at causing breathing problems. Radium occurs naturally in earth rock’s and is made primarily of alpha particles. 

During the process of nuclear decay, the liberated energy is shared between the daughter nucleus and the alpha particle. Alpha particles dissipate their energy during collisions by two mechanisms: electron and ionization excitation. The alpha particle with high charge is relative to other forms of nuclear radiation and gives greater ionization power.

Uses of Alpha Particles

• They are used as smoke detectors.

• They are commonly used in space probes

• They are also used in radiotherapy to treat cancer.

Beta Particles

Beta particles are electrons which are smaller than alpha particles. They can easily penetrate through human skin or cause tissue damage. Beta particles can be inhaled if they contaminate food and water supplies. Beta-decay is the production of beta particles. Beta particles denoted by Greek letters (β).

They normally occur in nuclei that have too many neutrons to achieve stability. They have a mass of half of one-thousandth of the mass of a proton. Their light mass means that they lose energy very quickly through interaction with matter. Beta particles are also found in the radioactive products of nuclear fission. They are also found in the radioactive chain of thorium, uranium, and actinium.

Uses of Beta Particles

• They are used in thickness detectors for the quality control of thin materials.

• Fluorine-18 is used as a tracer for PET.

• They also help in the treatment of eye and bone cancers.

• Tritium is used for emergency lighting.

Gamma Rays

It is a packet of electromagnetic energy emitted by the nucleus of some radioactive elements. Photons of gamma rays are the most energetic photons in the electromagnetic spectrum. They are basically emitted from an excited nucleus. 

Waves of gamma rays have the shortest wavelength. The high energy of gamma rays enables them to pass through many kinds of material including human tissues. Radiations of gamma rays are penetrating and interact with matter through ionization. 

They are also easily found in the radiation decay of thorium, uranium, etc. Gamma radiations are easily found in rocks, soil, and in our water and food.

Uses of Gamma Rays

• Cobalt-60 used in industrial radiography

• They are also used in pasteurization

• Caesium-137 used in measurement and control of the flow of liquid in industrial processes.

• They are also used in leveling gauges for packaging of food, and other products.

The rotational motion of the body is analogous to its translational motion. Also, the terms that are used in rotational motion such as the angular velocity and angular acceleration are analogous to the terms velocity and acceleration that are used in translational motion. Thus, we can say that the rotation of a body about a fixed axis is analogous to the linear motion of a body in translational motion. In this section, we will discuss the kinematics kinematic quantities in rotational motion like the angular displacement θ, angular velocity ω angular acceleration α respectively corresponding to kinematic quantities in translational motion like displacement x, velocity v and acceleration a.

Rotational Kinematics Equations

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Let us consider an object undergoing rotational motion about a fixed axis, as shown in the figure, and take a particle P on the rotating object for analyzing its motion. Now as the object rotates about the axis passing through O, the particle P gets displaced from one point to another, such that the angular displacement of the particle is θ.

If at time t = 0, the angular displacement of the particle P is 0 and at time t, its angular displacement is equal to θ, then the total will be θ in time interval t.

Similar to velocity, the rate of change of displacement of the angular velocity is the rate of change of angular displacement with time.

Mathematically, angular velocity,

w = dθ/dt

Further, Similar to acceleration that rate of change velocity the angular acceleration of the particle P is defined as the rate of change of angular velocity of the object wrt time.

Mathematically, angular acceleration,

α = dω/dt

Hence, we see that the kinematic quantities in the rotational motion of the object P are angular displacement(θ), the angular velocity(ω) and the angular acceleration(α) that corresponds to displacement(s), velocity(v) and acceleration(a) in linear or translational motion.

Kinematic Equations of Rotational Motion

We have already learned in the kinematics equations of linear or translational motion with uniform acceleration.

The three equation of motion was,

v = v0+ at

x = x0 + v0t + (1/2) at²

v² = v02+ 2ax

Where x0  is the initial displacement and v0 is the initial velocity of the particle.v and x are velocity and displacement respectively at any time t and is the constant acceleration throughout the linear motion. Here initial means t = 0. Now, this equation corresponds to the kinematics equation of the rotational motion as well because we saw above how the kinematics of rotational and translational motion was analogous to each other.

ω = ω0+ αt

θ = θ0 + ω0t + (1/2) αt²

ω² =  ω0² + 2α (θ – θ0)

Where  θ0  is the initial angular displacement of the rotating particle or body, ω0 is the initial angular velocity and α is the constant angular acceleration of the body while ω and θ is the angular velocity and displacement respectively at any time t after the start of motion.

We come across many days today as examples of the relation between the kinematics of rotating body and its translational motion, one of which is if a motorcycle wheel has a large angular acceleration for a fairly long time, it is spinning rapidly and rotates through many revolutions. Thus we can say that, if the angular acceleration of the wheel is large for a long period of time t, then the final angular velocity ω and angle of rotation θ are also very large. The rotational motion of the wheel is analogous to the motorcycle’s large transnational acceleration produces a large final velocity, and also the distance traveled will be large. Also, we can relate the angular displacement θ and translation displacement by equation

S = 2πrN

Where N is the number of a complete rotation of particle chosen at any point on the wheel 

N = θ/2π

An electronic LCR circuit contains a resistor of R ohms, a capacitor of C farad, and an inductor of L Henry, all connected in a series combination with each other. Since all the three elements of the LCR circuit are connected in series, the current passing through each of them is the same and is equivalent to the total current I passing through the circuit. A circuit that contains L, R, and C components at some particular frequencies can make the L and C (or some of their electrical effects) disappear completely. 

A Brief Explanation of LCR Circuit

The LCR circuit can act as just a capacitor, just a resistor, or just an inductor individually. The LCR circuit is also used to enhance the voltage to increase the voltage passing through the individual components of the circuit. 

This voltage can be much larger than the external voltage applied to the circuit. LCR circuits are also useful to change the impedance of the circuit, to increase or decrease the resistance to the current of different frequencies. All these effects can either be used separately or can be used all together to get the desired results in electronic devices. 

The three Components of an LCR circuit work together to Produce different Effects

Resistor: 

The resistor limits the current flow. It helps in controlling the power or voltage that is applied to the LCR circuit. The resistor is a component in an electronic device that limits the flow of electric current. The resistor helps control the amount of power or voltage that is applied to the LCR circuit. This is important because it prevents too much current from flowing through the other components in the circuit

Capacitor:

A capacitor stores energy and releases it in a controlled manner- It helps in controlling the voltage or power that is applied to the LCR circuit. The capacitor stores energy and releases it in a controlled manner, which prevents too much current from flowing through the L resistor.

Inductor:

An inductor resists change in current flow- It helps in controlling the fluctuations in current flow. The inductor resists change in current flow, which helps to stabilize the LCR circuit. The LCR circuit is used as a part of electronic devices such as cellphones, televisions, and computers to regulate the intensity of light emitted from these devices.

LCR Circuit Diagram

This diagram consists of all the components of the module, such as inductance, capacitance, and resistance. It fulfills along with its properties like Reactance, Impedance, and Phase.

This module discusses the overall effect of L, C, and R when connected in series and supplied by an alternating voltage. In such arrangements, the current provided passes through all the elements of the circuit equally. VR, VC, and VL symbolize the amount of individual voltage across the register, capacitor, and inductor, respectively.

There is some internal resistance on the applied voltage, which is measured across the inductor. In the LCR circuits, the internal and external resistance is usually there in the circuit. Therefore, it is easy to know that the voltage across VR is the total voltage across the circuit which inhibits the internal resistance L accompanied by a fixed resistor. Here [V_{s}] is the applied supplied voltage.

The phase relationship between the current of the circuit IS, and the supplied voltage VS depends on both, the relative values of the capacitance, inductance, and frequency of the applied voltage. Various conditions arise depending upon whether the inductive reactance [X_{L}] is smaller or higher than the capacitive reactance [X_{C}]. Diagrams can illustrate this.

As per the above diagram, one can infer that:

[ V^{2} = V_{R}^{2} + (V_{L} – V_{C}) ^{2} —(1)]

Since it is an LCR circuit, the equal current will pass through all components. Therefore,

[V_{C} = I_{R}] —(2)

[V_{L} = I X _{L}] —(3)

[V_{C} = I X _{C}]—-(4)

Using equation (1), (2), (3) and (4)

[ I = frac{V}{sqrt{R^{2} + (X_{L} – X_{C})^{2}}}]

The angle between I and V is known as phase shift,

[tan phi = frac{V_{L} – V_{C}}{R}],

In terms of impedance, it is represented as,

[tan phi = frac{X_{L} – X_{C}}{R}],

Three Possibilities Arise Depending upon the Values of [X _{C} and X_{L}]. 

1. If [X _{L}> X_{c}], then [tan phi > 0], in this case, the voltage leads the current, and the LCR circuit is said to be an inductive circuit.

2. If [X _{L} <  X_{c}], then [tan phi < 0], in this case, the current leads the voltage, and the LCR circuit is said to be a capacitive circuit.

3. If [X _{L} =  X_{c}], then [tan phi = 0], and the current is in phase with the voltage, and the circuit is known as a resonant circuit.

Overview

This module gives a brief introduction to some of the most beneficial and most creative circuits of the electronic world. The circuits are elementary, containing two or three components that are connected in series with each other. They perform various complex functions and have a broad range of circuit applications.

Electronic circuits are used to connect an indicator, a resistor, or a capacitor either in parallel or in series. Some previous modules of this series talk about the capacitors and inductors, and their connection with the resistors exclusively. This creates some useful circuits like filters, integrators, and differentiators.

Capacitors and Inductors have different purposes in an AC circuit. This module talks about the cumulative properties of reactance, the impedance of the capacitors, and the inductors with various frequencies to generate amazing effects.

Importance of LCR Circuit

LCR circuits are important in various applications. LCR circuits help reduce power consumption by controlling too much current flow through a device or component, causing it to overheat. LCR circuits also help reduce voltage fluctuations that can damage electronic devices.

Stores energy and releases it in a controlled manner which prevents too much current from flowing throughout the L resistor. It consists of three components L resistor, capacitor, and L inductor

Helps in controlling the fluctuations of current flow, which stabilizes the LCR circuit.

The momentum of an object is the product of the velocity and mass of an object. It is a vector quantity. Conservation of momentum is a fundamental law of physics, which states that the total momentum of an isolated system is conserved in the absence of an external force. In other words, the total momentum of a system of objects remains constant during any interaction if no external force acts on the system. The total momentum is the vector sum of individual momenta. Therefore, the component of the total momentum along any direction remains constant (whether the objects interact or not). Momentum remains conserved in any physical process.

Overview of the Law of Conservation of Momentum

Conservation of momentum states that the total momentum of an isolated system remains the same in the absence of an external force, i.e., the momentum can neither be created nor be destroyed, however, it can be changed through the action of forces as described by Newton’s laws of motion.

Momentum is the product of the mass of the object and the velocity at which it is travelling and is also equal to the total force required to bring the object to rest.

One of the real-life aspects of the conservation of momentum is collision problems in which the momentum remains conserved and the net external force remains zero.

Additionally, there are several applications of momentum conservation in our day-to-day life that we will cover on this page. Along with this, we will understand the logic behind this concept and the proof of the conservation of momentum.

Illustration in One-Dimension

Conservation of momentum can be explained through a one-dimensional collision of two objects. Two objects of masses m₁ and m₂ collide with each other while moving along a straight line with velocities u1 and u2, respectively.  After the collision, they acquire velocities v1 and v2 in the same direction.

Total momentum before collision pi=m1u1+m2u2

Total momentum after collision  pf=m1v1+m2v2

If no other force acts on the system of the two objects, total momentum remains conserved. Therefore,

Pi = pf

m1u1+m2u2= m1v1+m2v2

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Derivation of Conservation of Momentum

If no external force is exerted on the system of two colliding objects, the objects apply impulse on each other for a short interval of time at the point of contact. According to Newton’s third law of motion, the impulsive force applied by the first object on the second one is equal and opposite to the impulsive force applied by the second object on the first object.  

During the one-dimensional collision of two objects of masses m1 and m2, which have velocities u1 and u2 before collision and velocities v2 and v2 after the collision, the impulsive force on the first object is F21 (applied by the second object) and the impulsive force on the second object is F12 (applied by the first object). Applying Newton’s third law, these two impulsive forces are equal and opposite, i.e.,

F21 = − F12

If the time of contact is t,  the impulse of the force F21 is equal to the change in momentum of the first object. 

F21. t = m1v1 − m1u1

The impulse of force F12 is equal to the change in momentum of the second object.

F12. t = m2v2 − m2u2

From F21 = − F12

F21. t = − F12. t

m1v1 − m1u1 = − (m2v2 − m2u2)

m1u1+m2u2= m1v1+m2v2

This relation suggests that momentum is conserved during the collision.

Collision in Two – Dimensions

Before the collision, the total momentum is pix = p1 = m1v1, along the X – axis and piy = p2 = m2v2 along the Y – axis. After the collision, the total momentum is pfx = (m + M) ucosθ, along X-axis and pfy = (m+M)usinθ

Applying conservation of momentum,

pix = pfx

m1v1 = (m + M) ucosθ….(1)

piy = pfy

m2v2 = (m+M) usinθ…..(2)

Therefore, squaring and adding equations (1) and (2),

[(m_{1}v_{1})^{2} + (m_{2}v_{2})^{2} = (m+M)^{2} u^{2} (Cos^{2}Theta + Sin^{2}Theta )]

[u = frac{sqrt{m_{1}^{2}v_{1}^{2} + m_{2}^{2}v_{2}^{2}}}{(m + M)}]    

It is the speed of the combined object.

Dividing equation (2) by (1),

[tan Theta = frac{m_{2}v_{2}}{m_{1}v_{
1}}]

θ gives the direction of the velocity.

Conservation of Momentum Examples

• Recoil of a Gun: If a bullet is fired from a gun, both the bullet and the gun are initially at rest i.e. the total momentum before firing is zero. The bullet acquires a forward momentum when it gets fired. According to the conservation of momentum, the gun receives a backward momentum. The bullet of mass m is fired with forward velocity v. The gun of mass M acquires a backward velocity u. Before firing, the total momentum is zero so that the total momentum after firing is also zero.

0 = mv + Mu

u = -[frac{m}{M}]v

u is the recoil velocity of the gun. The mass of the bullet is much less than that of the gun i.e. m ≪ M. The backward velocity of the gun is very small,

u ≪ v

• Rocket Propulsion: Rockets have a gas chamber at one end, from which gas is ejected with enormous velocity. Before the ejection, the total momentum is zero. Due to the ejection of gas, the rocket gains a recoil velocity and acceleration in the opposite direction. This is a consequence of the conservation of momentum

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If a rocket of mass m ejects the propellent of small mass dm with an exhaust velocity ve such that the residual rocket of mass m – dm acquires a velocity dv in the opposite direction, the momenta of the propellant and the residual rocket are equal in magnitude and opposite in direction.

vedm = − (m − dm)dv

Since both  dm and dv are small, the equation can be approximated as

dv = − ve[frac{dm}{m}]

If the mass of the rocket reduces from mo to m’ as its velocity increases from 0 to v’, integrating the above equation

[int_{0}^{v^{1}}dv = -v_eint_{mo}^{m^{1}} dm/m]

v’ = ve ln[(frac{m}{mo})]

Solved Examples

I. A bullet of mass 6 g is fired with a speed of 500 m/s from a gun of mass 4 kg. What would be the recoil velocity of the gun?

Solution: The initial momenta of the bullet and the gun are zero such that the total initial momentum is zero. The bullet of mass m = 6g  is fired with forward velocity v = 500 m/s. The gun of mass M = 4kg acquires a backward velocity V. 

m = 6 g =[frac{6}{1000}]kg

According to the conservation of momentum formula,

0 = mv + MV

0 =[frac{6}{1000}]kg(500m/s) + (4kg) v

v = – 0.75 m/s

The recoil speed of the gun is 0.75 m/s. The negative sign implies that the recoil velocity is opposite to the velocity of the bullet.

A good explanation is worth a hundred readings, because if you have a clear explanation for the topic then chances are, you may not need to read the explanation again, because you learn, understand, and grasp everything in just one reading. And for the topic of Reflection of Light and Refraction of Light such explanation is very much needed. As it does not only help you in better understanding the topic of Reflection of Light and Refraction of Light, but it also helps you from lots of anxiety, saves lots of your time, and it boosts your morale.

But there is one more thing, which is, it becomes quite difficult for the students to find such an explanation. And hence for making the learning process of the students easy and for saving their time. has brought the complete explanation of Reflection of Light and Refraction of Light, in a language that is easy to understand and grasp for the students. And also, provides the complete explanation of the Reflection of Light and Refraction of Light completely free of cost.

Reflection of Light

• The process of sending back light rays that drop on an object’s surface is called Light reflection.

• Silver metal is also one of the best light reflectors.

• In home the mirrors we use on our dressing tables are plane mirrors.

• A ray of light is the straight line that the light travels along and a series of light rays is considered a light beam.

 Laws of Reflection of Light

• The angle of incidence at the point of incidence is equal to the angle of reflection and the incident radius, the reflected radius, and the normal mirror at the point of incidence.

• These laws apply to all types of reflective surfaces, including spherical surfaces

 

Characteristics of Images Formed by Mirrors

• Images created through mirrors are always virtual and erect

• Image size is always equal to the object size, and the image is inverted laterally.

• The images formed by the mirror on the plane are as far behind the mirror as the object facing the mirror.

• Lateral Inversion: If an object is placed in front of the mirror, the left side of the object tends to be the right side of this image. This transition in an object’s sides, and its mirror image, is called lateral inversion.

Spherical Mirrors

A circular mirror’s reflective surface may be angled inside or outwards.

There are two types of spherical mirrors 

1. Concave Mirror: – In a concave mirror light reflection occurs at the concave surface or bent-in surface as shown in the figure below.

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2. Convex Mirror: In a convex mirror the light is reflected on the convex surface or bent out as shown in the figure below

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Commonly Used Terms About Spherical Mirrors

1. Center of Curvature: – A spherical mirror reflecting form constitutes a part of a sphere. There is a center to this sphere. This point is termed the spherical mirror’s curvature center. The letter C is represented on it. Note that the curvature center isn’t a part of the mirror. This exists beyond its reflective surface. Before it lies the center of curvature of a concave mirror. However, in the case of a convex mirror, as shown above, it lies behind the mirror. 

2. Radius of Curvature: The angle of the sphere from which the reflecting surface of a spherical mirror forms a part, is considered the curvature radius of the mirror. The letter R is depicted on it.

3. Pole: A spherical mirror’s center is called its pole and is represented by the letter P as shown in the figure.

4. Principle Axis: The straight line that passes through the pole and the curvature center of a spherical mirror is called the mirror’s principal axis.

5. The Aperture of The Mirror: – Portion of the mirror from which the reflection of light actually occurs is called mirror aperture. The mirror opening actually represents mirror size.

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Overview of the Reflection of Light.

When the wavefront of the light returns into the medium from which it originally originated, it is called Reflection of Light. It happens because the wavefront changes its direction at an interface between the two different media. In simple language, we can say that when the lights get sent back from the surface of an object, upon which it lands, to the point of its origin is called reflection of the light. Here, the surface which throws the light back to its origin, that is to say, the surface which reflects the light, is called a reflector. 

Usually, the surfaces which have polished metal are good reflectors. Also, the mirror is one of the most common reflectors, especially out of those who are found in the household. Waxed surfaces and water surfaces also play the role of the reflector. But one of the best reflectors is the silver blaze.

Laws of Reflection of Light in Brief.

There are two laws of reflection of light, which are as under:

• First Law of Reflection: This law states that the reflected ray and the incidental ray all lie on the same plane.

• Second Law of Reflection: This law states that the angle of reflection and the angle of the incident are always going to be equal.

Overview of Refraction of Light.

For a long time, it was believed that the light travels in a straight line, but other theories regarding the light were developed in the last century, that is to say in the 20th century. And to a greater extent, these new theories help in developing and understanding the moment of light from one medium to another medium.

When light travels from one medium to another medium the direction of the propagation of light changes in another medium. To put it simply, when the light travels from one medium to another medium, its velocity or speed changes, and this change is called the refraction of light. The nature of another medium plays a good role in the refraction of light.

If you wish to learn more about the refraction of light, you may like to follow this link.

Principle Focus & Focal Length of Spherical Mirrors

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• From above figure we see a set of rays landing on a concave mirror parallel to the principal axis. Now, if we observe the reflected rays, we see that they all intersect on the mirror’s main axis at a point F. This feature is called the principal focus of the concave mirror.

• In the case of convex mirror rays, these reflected rays appear to originate from point F on the main axis and this point F is called the main focus of the convex mirror.

• The distance between the pole and a spherical mirror’s principal focus is called the focal length. The letter f is represented on it.

• There is a relationship between the curvature radius R and the focal length f of a spherical mirror and is given by R=2f, meaning that the main focus of a spherical mirror is between the pole and the curvature center.

 

Image Formation by Spherical Mirrors

• The existence, direction, and size of the image created by a concave mirror depend on the object’s position about points P, F, and C.

• The formed picture can be both actual and simulated, depending on the object’s position.

• The picture is magnified, diminished, or has the same dimension, depending on the object’s position.

 

Rules for Obtaining Images Formed by Spherical Mirrors

Rule 1

A ray of light parallel to the mirror’s principal axis passes through its focus after mirror reflection as shown in the figure below

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From the above figure, it can be clearly seen that the light rays in concave mirrors travel through the main focus and tend to differ from the main focus in concave mirrors.

Rule 2

A ray of light that passes through the curvature center of the concave mirror or is directed towards the curvature center of a convex mirror, is reflected back along the same path as shown in the figure below.

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Rule 3

A ray going through the main focus of a concave mirror or a ray that is directed towards the main focus of a convex mirror is after reflection parallel to the main axis and is shown in the figure below.

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Rule 4

A ray incidence is projected obliquely toward the main axis, toward a point P (mirror pole), on the concave mirror, or a convex mirror. The incident and reflected rays obey the reflecting rules at the point of incidence (point P), allowing equal angles to the main axis and shown in the diagram below

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Image Formation by Concave Mirror

1. Between pole P and focus F

2. At the focus

3. Between focus F and center of curvature C

4. At the center of curvature

5. Beyond the center of curvature

6. It is called infinity at far distances and cannot be shown in figures

• The picture formed by a concave mirror for the different object locations is shown in the table below

• Concave mirrors are used as spotlights, reflectors in car headlights, hand torches, and table lights.

• In the field of solar energy, large concave mirrors are used to focus sun rays on objects to be heated.

Image Formation by Convex Mirrors

• To create a ray diagram, we will have to follow the direction of light rays to figure out the position, shape and scale of the image created by the convex mirror.

• Upon reflection from the mirror, a beam of light parallel to the principal axis of a convex mirror appears to come from its center.

• A ray of light traveling to the center of convex mirror curvature is reflected back in its own direction.

• Convex mirrors have their focus and curvature center behind them and no light can go behind the convex mirror and all the rays we show behind the convex mirror are virtual and no ray actually passes through the concentration and curvature center of the convex mirror.

• Whatever the object’s position in front of the convex mirror, the convex mirror image is always behind the mirror, virtual, erect, and smaller than the object.

• In the table below is the existence, position, and relative size of the image created by a convex mirror

• Convex mirrors are used in automobiles as rear-view mirrors to see the traffic on the backside as they give erect images and also a highly decreased one that gives the wide-field view of traffic behind.

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Sign Convention for Reflection by Spherical Mirrors

Spherical mirrors reflect light following a set of sign conventions called the New Cartesian Sign Convention. In this convention, the mirror’s pole (P) is taken as the root. The mirror’s principal axis is taken as the coordinate system’s x-axis (X’X). The following are the Conventions

• The object is always situated to the mirror’s left. This implies that the light from the object falls on the left side of the mirror.

• All distances are measured from the mirror pole parallel to the principal axis.

• All distances measured to the right of the origin (along + x-axis) are taken as positive while those measured to the left of the origin are taken as negative (along-x-axis).

• Positives are taken distances measured perpendicular to and above the main axis (along the y-axis).

• Distances determined perpendicularly to and below the main axis (along -y-axis) are considered negative.

The figure below shows these new Cartesian sign conventions for spherical mirrors

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Lorentz transformation refers to the relationship between two coordinate frames that move at a constant speed and are relative to one another. It is named after a Dutch physicist, Hendrik Lorentz.

We can divide reference frames into two categories:

• Frames of inertial motion – motion with a constant velocity

• Non-inertial Frames – Rotational motion with constant angular velocity and acceleration in curved paths

Lorentz Transformation in Inertial Frame

A Lorentz transformation can only be used in the context of inertial frames, so it is usually a special relativity transformation. During the linear transformation, a mapping occurs between 2 modules that include vector spaces. The multiplication and addition operations on scalars are preserved when using a linear transformation. As a result of this transformation, the observer who is moving at different speeds will be able to measure different elapsed times, different distances, and order of events, but it is important to follow the condition that the speed of light should be equivalent across all frames of reference.

Lorentz Boost

It is also possible to apply the Lorentz transform to rotate space. A rotation free of this transformation is called Lorentz boost. This transformation preserves the space-time interval between two events.

The Statement of the Principle

The transformation equations of Hendrik Lorentz relate two different coordinate systems in an inertial reference frame. There are two laws behind Lorentz transformations:

• Relativity Principle

• Light’s constant speed

Simplest Derivation of Lorentz Transformation

We will start by scaling Galilean transformations by Lorentz factor (γ) which is-

γ = [frac{1}{sqrt{1 – frac{v^{2}}{c^{2}}}}]

γ = [frac{1}{sqrt{1 – β^{2}}}]

Galilean transformations of Newtonian transformations: –

t’=t

z’=z

y’=y

x’=x- vt

Here, x’,  y’ , z’ and ct’  are the new coordinates. We need to transform from x to x’ and ct to ct’.

This implies, x’ = γ(x – βct)

And, ct’ =  γ(ct – βx)

Extending it to 4 dimensions,

y’=y

z’=z

Another form of writing the equations, is to substitute β = [frac{v}{c}]

γ = [frac{1}{sqrt{1 – frac{v^{2}}{c^{2}}}}] = [frac{1}{sqrt{1 – β^{2}}}]

x’ = γ(x – ct[frac{v}{c}])

x’ = γ(x – vt)

ct’ =  γ(ct – βx)

ct’ =  γ(ct – [frac{v}{c}]x)

Dividing by c ,

[frac{ct’}{c}] = γ([frac{ct}{c}] – [frac{vx}{c^{2}}])

t’ =  γ(t – [frac{vx}{c^{2}}])

When , v << c , Then [frac{vx}{c^{2}}] ≈ 0

 and when γ is equal to 1,

t’ = γ(t – [frac{vx}{c^{2}}]) becomes t’ ≈ t

x’ = γ(x – vt) becomes x’ = x – vt

 

Equation of the Lorentz Transformation

Lorentz transformations transform one frame of spacetime coordinates into another frame that moves at a constant speed relative to the other. The four axes of spacetime coordinate systems are x, ct, y, and z.

x’ = γ(x – βct)

ct’ =  γ(ct – βx)

Extending it to 4 dimensions,

y’=y

z’=z

Space-Time

The concept of Lorentz transformation requires us to first understand spacetime and its coordinate system.

As opposed to three-dimensional coordinate systems having x, y, and z axes, space-time coordinates specify both space and time (four-dimensional coordinate system). The coordinates of each point in four-dimensional spacetime consist of three spatial and one temporal characteristic.

Need of a Spacetime Coordinate System

Earlier, time was viewed as an absolute quantity. Since space is not an absolute quantity, observers would disagree about the distance (thus, the observers would not agree about the speed of the light) even though they agree on the time it takes for the light to travel. 

Consequently, time is no longer considered an absolute quantity due to the Theory of Relativity.

As a result, the distance between events can now be calculated as a function of time. 

d = (1/2)c

Where,  

The theory of relativity has changed our understanding of space and time as separate and independent components. Therefore, space and time had to be combined into one continuum.

World-Line

The path that an object follows as it moves through a spacetime diagram is called its world line. Spacetime diagrams are important because world lines may not correspond to paths that objects traverse in space. For example, when a car moves with uniform acceleration, the graph in a velocity-time graph is no longer a straight line. In your reference frame, a world line is a stationary straight line whose x coordinate is always equal to zero.

Fun Facts about Lorentz Transformation

1. The world line of the speed of light is the only such path that does not change when followed by a series of contraction and expansion.

2. The world line of the speed of light is always at an angle of 45° to the spacetime coordinate system.