# 11th Standard - Physics Tutorial - NCERT - CBSE Pattern

We will cover below chapters.- 1 - UNITS AND MEASUREMENTS
- 2 - MOTION IN A STRAIGHT LINE
- 3 - MOTION IN A PLANE
- 4 - LAWS OF MOTION
- 5 - WORK, ENERGY AND POWER
- 6 - SYSTEM OF PARTICLES AND ROTATIONAL MOTION
- 7 - GRAVITATION
- 8 - Mechanical Properties of Solids
- 9 - Mechanical Properties of Fluids
- 10 - Thermal Properties of Matter
- 11 - Thermodynamics
- 12 - Kinetic Theory
- 13 - Oscillations
- 14 - Waves

# UNITS AND MEASUREMENTS

## 1.1 Introduction

## 1.2 The international system of units

Standard units are length, time, mass, electric current, thermodynamic temperature, amount of substance, and luminous intensity. The units (e.g. metre, second, kilogram) for the fundamental or base quantities are called fundamental or base units. There are lot of other units which can be derived from base units. e.g. Frequency's unit is hertz which has been derived by taking reciprocal of second (also called as Inverse Second) ! Other examples of derived units are joule, newton, watt. Base units and derived units together are called as system of units. When taking measurements, errors may occur in readings. It's important to reduce the error rate as much as possible.## 1.3 Significant figures

In measured and computed quantities proper significant figures only should be retained. Rules for determining the number of significant figures, carrying out arithmetic operations with them, and rounding off the uncertain digits must be followed.## 1.4 Dimensions of physical quantities

time (T), length (L), mass (M), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J) are standard dimensions.## 1.5 Dimensional formulae and dimensional equations

## 1.6 Dimensional analysis and its applications

Dimensional analysis can be used to check the dimensional consistency of equations, deducing relations among the physical quantities.# MOTION IN A STRAIGHT LINE

## 2.1 Introduction

Rectilineaer motion is the study of motion of objects along a straight line. In Kinematics, we study motion without taking into consideration the causes of motion.## 2.2 Instantaneous velocity and speed

## 2.3 Acceleration

## 2.4 Kinematic equations for uniformly accelerated motion

# MOTION IN A PLANE

## 3.1 Introduction

## 3.2 Scalars and vectors

If Vector a is at an angle of $\theta$, then we can split it into 2 components.x component $a cos\theta$

y component $a sin\theta$

## 3.3 Multiplication of vectors by real numbers

## 3.4 Addition and subtraction of vectors - graphical method

Vectors can be added in a simple way if it is given in the form of i,j,k format.## 3.5 Resolution of vectors

## 3.6 Vector addition - analytical method

When magnitude and angle between 2 vectors a and b is given, we can add vectors by using below 2 approaches. When 2 vectors are added, we get Resultant Vector!- $\sqrt { a^2 + b^2 + 2ab \cdot cos\theta}$ and we can also get $tan \alpha = \frac { b sin\theta} {a + bcos\theta }$
- This is called as triangle law of addition or parallelogram law of addition

## 3.7 Motion in a plane

When motion happens in x as well as y direction, it is called as a motion in plane!## 3.8 Motion in a plane with constant acceleration

## 3.9 Projectile motion

We can easily calculate any parameters by applying rectilinear equations of motion for horizontal and vertical components of vector. If Vector a is at an angle of $\theta$, then we can split it into 2 components.x component $a cos\theta$

y component $a sin\theta$

So maximum height of object thrown at an angle of theta and with speed of v will be

$h_{max} = \frac {v_y^2} {2a}$

$h_{max} = \frac {v^2 \cdot sin^2\theta} {2a}$

Time of flight will be $T_{flight} = \frac {2vsin\theta} {g}$

Range of flight will be $R_{flight} = \frac {v^2sin2\theta} {g}$

## 3.10 Uniform circular motion

### Linear and Angular Relation

We know below relation which states that angular displacement is equal to product of arc length and radius of circle.# LAWS OF MOTION

## 4.1 Introduction

## 4.2 Aristotle's fallacy

Aristotle's view that a force is necessary to keep a body in uniform motion is wrong. A force is necessary in practice to counter the opposing force of friction.## 4.3 The law of inertia

## 4.4 Newton's first law of motion

Galileo extrapolated simple observations on motion of bodies on inclined planes, and arrived at the law of inertia. Object will maintain it's state (motion or rest) until external force is applied! In simple terms, the First Law is "If external force on a body is zero, its acceleration is zero".## 4.5 Newton's second law of motion

Momentum (P) = mass x velocity- The second law is consistent with the First Law (F = 0 implies a = 0)
- It is a vector equation
- It is applicable to a particle, and also to a body or a system of particles, provided F is the total external force on the system and a is the acceleration of the system as a whole.
- F at a point at a certain instant determines a at the same point at that instant. That is the Second Law is a local law; a at an instant does not depend on the history of motion.

## 4.6 Newton's third law of motion

Every action has equal and opposite reaction! Forces in nature always occur between pairs of bodies. Force on a body A by body B is equal and opposite to the force on the body B by A. Action and reaction forces are simultaneous forces. There is no cause-effect relation between action and reaction. Any of the two mutual forces can be called action and the other reaction. Action and reaction act on different bodies and so they cannot be cancelled out. The internal action and reaction forces between different parts of a body do, however, sum to zero.## Tips to solve the problems

Ideal string is massless and inextensible. But in real life, strings can have a mass and they can be extensible. If string is massless, Tension is constant between string of 2 blocks. If string has mass, tension is not constant. Below types of problems can be asked.- Block in horizontal(e.g. on table - Normal Force and gravitational force cancel out) and vertical position (hanging - gravitational force and Tension force involved here. No normal force!!)
- Blocks on inclined surface - Split the force in vertical and horizontal components
- Multiple blocks connected by string (Tension force) in horizontal and vertical position - watch https://www.youtube.com/watch?v=Anmu6TXSvCw
- Blocks on surface with specific coefficient of static and kinetic friction (Frictional Force)
- String or rope connected to hooks
- Pully
- Object inside Lift/Elevator
- Lift Rope Tension
- Pendulum in lift
- Object in circular motion
- Stack of blocks
- Object moving in rotational motion - Watch pendulum video
- Torque problems
- Object inside Lift from non inertial frame (psuedo force)
- Combination of all of above

- Draw FBD
- If system is stationary, Net force is 0
- If system is moving, Net force is the product of mass and acceleration. Net force can be calculated by considering total mass of entire system (e.g. all blocks).
- Direction of acceleration is considered as positive

- Force , mass, acceleration, time relationship
- Time and force to move object of mass m by 1 meter
- Force and displacement relationship

## 4.7 Conservation of momentum

The total momentum of an isolated system of particles is conserved. The law follows from the second and third law of motion. Momentum of 2 bodies is conserved.## 4.8 Equilibrium of a particle

When sum of all forces from all directions is 0, body is said to be in an equilibrium! When a body is on inclined surface in equilibrium, then below rules apply.## 4.9 Common forces in mechanics

here is the list of common forces.- Tension - If an object of mass m is hanging from the rope, tension will be equal to weight of object (mg) provided there is no acceleration in object.
- Spring - Spring Force is given by below formula.$force = kx$where k is spring constant and x is the length by which spring is stretched.
- Contact
- Normal
- Friction - Static , limiting and Kinetic (sliding)
- Rolling

Frictional force opposes (impending or actual) relative motion between two surfaces in contact. It is the component of the contact force along the common tangent to the surface in contact. Static friction fs opposes impending relative motion; kinetic friction fk opposes actual relative motion. They are independent of the area of contact and satisfy the following approximate laws. - Centripetal

## 4.10 Circular motion

## 4.11 Solving problems in mechanics

# WORK, ENERGY AND POWER

## 5.1 Introduction

## 5.2 Notions of work and kinetic energy : The work-energy theorem

Change in the kinetic energy is the work done by the net force! This is called as work energy theorem!## 5.3 Work

Work is 0 if- Displacement is 0
- or force is 0
- or the force and displacement are mutually perpendicular

## 5.4 Kinetic energy

## 5.5 Work done by a variable force

## 5.6 The work-energy theorem for a variable force

If we have a force - displacement graph, the work done is given by the area under graph!## 5.7 The concept of potential energy

A force is conservative if (i) work done by it on an object is path independent and depends only on the end points ${xi, xj}$, or (ii) the work done by the force is zero for an arbitrary closed path taken by the object such that it returns to its initial position!## 5.8 The conservation of mechanical energy

The principle of conservation of mechanical energy states that the total mechanical energy of a body remains constant if the only forces that act on the body are conservative. The gravitational potential energy of a particle of mass m at a height x about the earth’s surface is V(x) = m g x## 5.9 The potential energy of a spring

The elastic potential energy of a spring of force constant k and extension x is## 5.10 Power

## 5.11 Collisions

# SYSTEM OF PARTICLES AND ROTATIONAL MOTION

## 6.1 Introduction

A rigid body is one for which the distances between different particles of the body do not change, even though there are forces on them. A rigid body fixed at one point or along a line can have only rotational motion. A rigid body not fixed in some way can have either pure translational motion or a combination of translational and rotational motions. In rotation about a fixed axis, every particle of the rigid body moves in a circle which lies in a plane perpendicular to the axis and has its centre on the axis. Every Point in the rotating rigid body has the same angular velocity at any instant of time. In pure translation, every particle of the body moves with the same velocity at any instant of time. Angular velocity is a vector. Its magnitude is $ω = \frac {d\theta} {dt}$ and it is directed along the axis of rotation. For rotation about a fixed axis, this vector ω has a fixed direction. The vector or cross product of two vector a and b is a vector written as $a \times b$. The magnitude of this vector is $absinθ$ and its direction is given by the right handed screw or the right hand rule. The linear velocity of a particle of a rigid body rotating about a fixed axis is given by$v = ω \times r$ , where r is the position vector of the particle with respect to an origin along the fixed axis. The relation applies even to more general rotation of a rigid body with one point fixed. In that case r is the position vector of the particle with respect to the fixed point taken as the origin.## 6.2 Centre of mass

The centre of mass of a system of n particles is defined as the point whose position vector is## 6.3 Motion of centre of mass

Velocity of the center of mass of a system of particles is given by $V = \frac {P} {M}$ , where P is the linear momentum of the system. The centre of mass moves as if all the mass of the system is concentrated at this point and all the external forces act at it. If the total external force on the system is zero, then the total linear momentum of the system is constant.## 6.4 Linear momentum of a system of particles

## 6.5 Vector product of two vectors

## 6.6 Angular velocity and its relation with linear velocity

## 6.7 Torque and angular momentum

The turning effect of force is called as Torque ($\tau$).Relation between torque and angular momentum is

The angular momentum of a system of n particles about the origin is

## 6.8 Equilibrium of a rigid body

A rigid body is in mechanical equilibrium if- it is in translational equilibrium, i.e., the total external force on it is zero $\sum F_i = 0$ and
- it is in rotational equilibrium, i.e. the total external torque on it is zero :$\sum \tau_i = \sum r_i \times F_i = 0$

## 6.9 Moment of inertia

Like inertia in translational motion, there is a moment of inertia in rotational motion. Property of body to resist angular change is called as moment of inertia!- Mass of body
- Mass distribution
- Size and shape
- Axis of rotation

- Find center of mass. So you will know distance between particle and center of mass
- Use moment of inertia formula

The moment of intertia of a rigid body about an axis is defined by the formula

Perpendicular axis theorem

Parallel axis theorem

### Moment of inertia of different shapes

You need to know below formulaeQuantity | Linear Motion | Rotation Motion |
---|---|---|

Displacement | $\Delta x$ | $\Delta \theta$ |

Speed | $v$ | $\omega$ |

Acceleration | $a$ | $\alpha$ |

Inertia | $m$ | $I$ |

Kinetic Energy | $K = \frac {1}{2} m v ^2$ | $K = \frac {1}{2} I \omega^2$ |

Momentum | $P = mv$ | $L = I \omega$ |

Force - Change in Momentum | $F = \frac{dP}{dt}$ $F = ma$ | $\tau = \frac{dL}{dt}$ $\tau = I \alpha$ |

Work | $W = F \Delta x$ | $W = \tau \Delta \theta$ |

Impulse | $F \Delta t$ | $\tau \Delta t$ |

## 6.10 Kinematics of rotational motion about a fixed axis

The kinetic energy of rotation is## 6.11 Dynamics of rotational motion about a fixed axis

## 6.12 Angular momentum in case of rotations about a fixed axis

# GRAVITATION

## 7.1 Introduction

Newton's law of universal gravitation states that the gravitational force of attraction between any two particles of masses m1 and m2 separated by a distance r has the magnitude.Below formula gives the gravitational force on object with mass m at a distance of r from the center of the Earth. M is the mass of earth and R is the radius of the Earth.

## 7.2 Kepler's laws

Kepler's laws of planetary motion state that- All planets move in elliptical orbits with the Sun at one of the focal points
- The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals. This follows from the fact that the force of gravitation on the planet is central and hence angular momentum is conserved.
- The square of the orbital period of a planet is proportional to the cube of the semi-major axis of the elliptical orbit of the planet The period T and radius R of the circular orbit of a planet about the Sun are related by

## 7.3 Universal law of gravitation

Gravitational field at point located at a distance of x from the uniform ring is given by below formula.## 7.4 The gravitational constant

## 7.5 Acceleration due to gravity of the earth

Acceleration due to gravity of earth is given by below formula.## 7.6 Acceleration due to gravity below and above the surface of earth

The acceleration due to gravity.(a) at a height h above the earth's surface

## 7.7 Gravitational potential energy

The gravitational force is a conservative force, and therefore a potential energy function can be defined. The gravitational potential energy associated with two particles separated by a distance r is given by## 7.8 Escape speed

The escape speed from the surface of the earth is## 7.9 Earth satellites

## 7.10 Energy of an orbiting satellite

# Mechanical Properties of Solids

## 8.1 Introduction

## 8.2 Stress and strain

Stress is the restoring force per unit area and strain is the fractional change in dimension! 3 types of stress- Tensile Stress or Compressive Stress
- Shearing Stress
- Hydraulic Stress

## 8.3 Hooke's law

Hook's law says that "For small deformations, stress is directly proportional to strain!"- Young's Modulus
- Shear Modulus
- Bulk Modulus

## 8.4 Stress-strain curve

## 8.5 Elastic moduli

### Young's Modulus

When an object is under tension or compression, the Hooke's law is given by formula F/A = YΔL/L where ΔL/L is the tensile or compressive strain of the object, F is the magnitude of the applied force causing the strain, A is the cross-sectional area over which F is applied (perpendicular to A) and Y is the Young's modulus for the object. The stress is F/A.### Shear Modulus

A pair of forces when applied parallel to the upper and lower faces, the solid deforms so that the upper face moves sideways with respect to the lower. The horizontal displacement ΔL of the upper face is perpendicular to the vertical height L. This type of deformation is called shear and the corresponding stress is the shearing stress. In this kind of deformation the Hooke's law is given by formula F/A = G x ΔL/L where ΔL is the displacement of one end of object in the direction of the applied force F, and G is the shear modulus.### Bulk Modulus

When an object undergoes hydraulic compression due to a stress exerted by a surrounding fluid, the Hooke's law is given by formula p = B (ΔV/V), where p is the pressure (hydraulic stress) on the object due to the fluid, ΔV/V (the volume strain) is the absolute fractional change in the object's volume due to that pressure and B is the bulk modulus of the object.## 8.6 Applications of elastic behaviour of materials

# Mechanical Properties of Fluids

## 9.1 Introduction

A liquid is incompressible and has a free surface of its own. Liquids are often considered to be incompressible because they have very little volume change in response to pressure. This is a result of the intermolecular forces and the arrangement of particles in a liquid. A gas is compressible and it expands to occupy all the space available to it## 9.2 Pressure

If F is the normal force exerted by a fluid on an area A then the average pressure Pav is defined as the ratio of the force to area## 9.3 Streamline flow

The volume of an incompressible fluid passing any point every second in a pipe of non uniform crossection is the same in the steady flow. v A = constant ( v is the velocity and A is the area of crossection) The equation is due to mass conservation in incompressible fluid flow.Nature of flow is deteremined by Reynolds number!

- If $R < 1000$, flow is steady
- If $1000 < R < 2000$ , flow is unsteady
- If $R > 2000$, flow is turbulent

## 9.4 Bernoulli's principle

Bernoulli's principle states that as we move along a streamline, the sum of the pressure (P), the kinetic energy per unit volume ( $\rho \frac {v^2}{2}$) and the potential energy per unit volume (ρgy) remains a constant.## 9.5 Viscosity

The ratio of the shear stress to the time rate of shearing strain is known as coefficient of viscosity - η. Stokes' law states that the viscous drag force F on a sphere of radius a moving with velocity v through a fluid of viscosity is## 9.6 Surface tension

Surface tension is a force per unit length (or surface energy per unit area) acting in the plane of interface between the liquid and the bounding surface. Molecules have more energy on the surface!# Thermal Properties of Matter

## 10.1 Introduction

## 10.2 Temperature and heat

## 10.3 Measurement of temperature

Thermometer can be used to measure the temperature. Thermometer uses thermometric property! Celcius temperature can be converted into Farenheit temperature using below formula.## 10.4 Ideal-gas equation and absolute temperature

The ideal gas equation is given by below formula.## 10.5 Thermal expansion

The coefficient of linear expansion ($α_l$ ) and volume expansion ($α_v$ ) are defined by below formula.## 10.6 Specific heat capacity

Specific heat capacity of a substance is given by below formula.## 10.7 Calorimetry

The latent heat of fusion ($L_f$ ) is the heat per unit mass required to change a substance from solid into liquid at the same temperature and pressure. The latent heat of vaporisation ($L_v$ )is the heat per unit mass required to change a substance from liquid to the vapour state without change in the temperature and pressure.## 10.8 Change of state

## 10.9 Heat transfer

Three modes of heat transfer are- conduction
- convection
- radiation

## 10.10 Newton's law of cooling

Newton's Law of Cooling says that the rate of cooling of a body is proportional to the excess temperature of the body over the surroundings.# Thermodynamics

## 11.1 Introduction

## 11.2 Thermal equilibrium

## 11.3 Zeroth law of thermodynamics

The zeroth law of thermodynamics states that "two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other". The Zeroth Law leads to the concept of temperature.## 11.4 Heat, internal energy and work

Internal energy of a system is the sum of kinetic energies and potential energies of the molecular constituents of the system. It does not include the over-all kinetic energy of the system. Heat and work are two modes of energy transfer to the system. Heat is the energy transfer arising due to temperature difference between the system and the surroundings. Work is energy transfer brought about by other means, such as moving the piston of a cylinder containing the gas, by raising or lowering some weight connected to it. Internal energy (U) is given by below formula!Degree of freedom is given below

- Monoatomic gas - 3
- Diatomic gas - 5
- Triatomic Linear - 5
- Triatomic Non Linear - 6 and more
- Polyatomic - 6 and more

## 11.5 First law of thermodynamics

The first law of thermodynamics is the general law of conservation of energy applied to any system in which energy transfer from or to the surroundings (through heat and work) is taken into account. Formula is## 11.6 Specific heat capacity

Specific heat capacity of a substance is given by below formula.## 11.7 Thermodynamic state variables and equation of state

Equilibrium states of a thermodynamic system are described by state variables. The value of a state variable depends only on the particular state, not on the path used to arrive at that state. Examples of state variables are pressure (P ), volume (V ), temperature (T ), and mass (m ). Heat and work are not state variables. An Equation of State (like the ideal gas equation PV = μ RT ) is a relation connecting different state variables.## 11.8 Thermodynamic processes

A quasi-static process is an infinitely slow process such that the system remains in thermal and mechanical equilibrium with the surroundings throughout. In a quasi-static process, the pressure and temperature of the environment can differ from those of the system only infinitesimally. In an isothermal expansion of an ideal gas from volume V1 to V2 at temperature T the heat absorbed (Q) equals the work done (W ) by the gas, each given by## 11.9 Second law of thermodynamics

The second law of thermodynamics disallows some processes consistent with the First Law of Thermodynamics. It statesKelvin-Planck statement - "No process is possible whose sole result is the absorption of heat from a reservoir and complete conversion of the heat into work."

Clausius statement - "No process is possible whose sole result is the transfer of heat from a colder object to a hotter object."

The Second Law implies that no heat engine can have efficiency η equal to 1 or no refrigerator can have co-efficient of performance α equal to infinity.

## 11.10 Reversible and irreversible processes

A process is reversible if it can be reversed such that both the system and the surroundings return to their original states, with no other change anywhere else in the universe. Spontaneous processes of nature are irreversible. The idealised reversible process is a quasi-static process with no dissipative factors such as friction, viscosity, etc## 11.11 Carnot engine

Carnot engine is a reversible engine operating between two temperatures T1 (source) and T2 (sink). The Carnot cycle consists of two isothermal processes connected by two adiabatic processes. The efficiency of a Carnot engine is given by- If $Q > 0$ , heat is added to the system
- If $Q < 0$, heat is removed to the system
- If $W > 0$, Work is done by the system
- If $W < 0$, Work is done on the system

# Kinetic Theory

## 12.1 Introduction

## 12.2 Molecular nature of matter

## 12.3 Behaviour of gases

## 12.4 Kinetic theory of an ideal gas

The ideal gas equation connecting pressure (P), volume (V) and absolute temperature (T ) is## 12.5 Law of equipartition of energy

The translational kinetic energy## 12.6 Specific heat capacity

## 12.7 Mean free path

# Oscillations

## 13.1 Introduction

## 13.2 Periodic and oscilatory motions

All Oscillatory motions are periodic but not vice versa. e.g. To and Fro motion and pendulum are oscillatory motions as well as periodic motion.## 13.3 Simple harmonic motion

Important terms- Amplitude A - Max displacement from mean point
- Time Period T - Time to complete one cycle
- Frequency f - Number of cycles per second
- Angular Frequency - Number of radians per second $\omega = \frac {2\pi} {T}$ This can also be written as $\omega = {2\pi} \times {Frequency}$
- Phase - Position and direction of particle at specific point

## 13.4 Simple harmonic motion and uniform circular motion

Restoring force is directly proportional to the displacement.## 13.5 Velocity and acceleration in simple harmonic motion

## 13.6 Force law for simple harmonic motion

## 13.7 Energy in simple harmonic motion

## 13.8 The Simple Pendulum

Any pendulum undergoes simple harmonic motion when the amplitude of oscillation is small.# Waves

## 14.1 Introduction

3 types of waves- Mechanical waves - need medium e.g. sound waves, waves on a string, water waves, sound waves, seismic waves
- Electromagnetic waves - may not need medium (Medium is optional). But can travel via medium as well!
- Matter waves - need medium - e.g. De-broglie wave in Quantum Mechanics

## 14.2 Transverse and longitudinal waves

In transverse waves, oscillations happen perpendicular to the direction of propogation of wave. In longitudinal waves, oscillations happen in the same direction of propogation of wave.Equation of wave going in left direction

Maximum particle velocity is at

## 14.3 Displacement relation in a progressive wave

## 14.4 The speed of a travelling wave

## 14.5 The principle of superposition of waves

## 14.6 Reflection of waves

## 14.7 Beats

# Electric and Magnetic field

Quantity | Electric Field | Magnetic Field |
---|---|---|

Symbol | $E$ | $B$ |

Unit | volt / meter | Tesla (T) or Gauss (G) |

Field creator | Charge (Coloumb) | Pole strength (Ampere meter) |

Field | $E = (\frac {1} { 4 \pi \epsilon_0 }) \frac { q} { r^2}$ where $\epsilon_0$ is Vacuum permittivity. | $B = (\frac {\mu_0} { 4 \pi }) \frac { m} { r^2}$ where $\mu_0$ is The vacuum magnetic permeability, also known as the magnetic constant, is the magnetic permeability in a classical vacuum. |

Force | $F_E = (\frac {1} { 4 \pi \epsilon_0 }) \frac { q_1 q_2} { r^2}$ | $F_B = (\frac {\mu_0} { 4 \pi }) \frac { m_1 m_2} { r^2}$ |

Dipole Moment | $P = q r$ | $M = m r$ |

Field for Dipole ( $l << r$ ) | $E_{axis} = (\frac {1} { 4 \pi \epsilon_0 }) \frac { 2P} { r^3}$ $E_{equator} = (\frac {1} { 4 \pi \epsilon_0 }) \frac { P} { r^3}$ | $B_{axis} = (\frac {\mu_0} { 2 \pi }) \frac { M} { r^3}$ $B_{equator} = (\frac {\mu_0} { 4 \pi }) \frac { M} { r^3}$ |

Flux | Unit of Electric flux is volt meter! $\phi_E = E \cdot S = E S cos \theta$ | Unit of Magnetic flux is Weber or $Tesla \text{ } {meter}^2$ . Magnetic flux density of one wb per meter square is 1 tesla. $\phi_B = B \cdot S = B S cos \theta$ |

# Modern Physics

## Atom

Particle | Charge | Mass |
---|---|---|

Proton | $+1.6 \times 10^ {-19}$ | $1.67 \times 10^ {-27}$ |

Electron | $-1.6 \times 10^ {-19}$ | $9.11 \times 10^ {-31}$ |

Newtron | No charge | $1.67 \times 10^ {-27}$ |

## De Broglie Wavelength

Louis de Broglie assumed that for particles (like electron) the same relations are valid as for the photon.The relationship between electric potential (also known as electric potential energy) and the electron wavelength is described by the de Broglie wavelength of electrons, which is a fundamental concept in quantum mechanics.

The de Broglie wavelength ($\lambda$) of a particle, such as an electron, is given by:

Where:

- $\lambda$ is the de Broglie wavelength.
- $h$ is Planck's constant.
- $p$ is the momentum of the particle.

Now, let's consider the relationship between electric potential and the de Broglie wavelength of an electron. When an electron is in an electric field and is subjected to an electric potential difference (voltage), it gains kinetic energy due to the electric field. The kinetic energy $K$ of the electron can be related to its charge $q$ and the electric potential difference ($V$ ) through the equation:

In quantum mechanics, the momentum of a particle is related to its kinetic energy ( K ) by the equation:

Where:

- $p$ is the momentum of the electron.
- $m$ is the mass of the electron.

Combining these equations, we can express the de Broglie wavelength of the electron in terms of its charge, the electric potential difference, and its mass:

This equation relates the electron's de Broglie wavelength to the electric potential ( V ) it experiences. It shows that as the electron gains more kinetic energy (due to a higher electric potential difference), its de Broglie wavelength will change accordingly.