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ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS Essay Example

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25ELECTROMAGNETIC THEORY

Electromagnetic Theory and Plasma Physics & Thermal and Statistical Physics

Name:

Institution:

Electromagnetic Theory and Plasma Physics & Thermal and Statistical Physics

Electromagnetic Theory and Plasma Physics

Home Assignment 1

Problem 1

  1. For spherical the angle the z-axis and the radius vector that connects the origin to the point of consideration is, θ. The angle between the projection of the radius vector onto the x-y plane and the x-axis is φ.

Therefore for the function in spherical coordinate system:

The spherical coordinates (r, θ,φ)

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 1

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 2

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 3

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 4

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 5

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 6

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 7

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 8

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 9

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 10

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 11

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 12

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 13

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 14

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 15

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 16

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 17

  1. For a cylindrical coordinate system:

Cylindrical coordinates (ρ, φ, z)

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 18

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 19

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 20

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 21

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 22

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 23

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 24

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 25

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 26

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 27

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 28

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 29

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 30

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 31

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 32

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 33

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 34

Problem 2

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 35

Since there is full spherical symmetry, the derivatives with respect to ? and ϕ must be zero

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 36

Total charge is Q = ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 37

Using the Laplace law: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 38

Solution of form: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 39

The zero potential is arbitrary and choosing the zero potential at infinity for the localized charges the value of b = 0 and the sphere charge looks like a point charge at large distances.

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 40

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 41

Using the form

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 42

Substituting in the poison’s equation it gives:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 43

Giving ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 44

Using boundary conditions at the surface of the sphere: r = R

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 45

Thus giving: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 46

The potential inside the sphere is given as:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 47

Thus the total charge is ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 48qsinELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 49

Problem 3

  1. Evaluate

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 50

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 51

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 52

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 53

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 54

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 55

  1. ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 56

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 57

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 58 = 0

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 59

Problem 4

An expression for volume charge density ? (r) of a point charge q at r’

The charge density can be expressed as the amount of electric charge in a volume.

For continuous charges, the integral of the charge density is given as:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 60

This relation explains how the charge density varies in many dimensions.

For a homogenous charge density:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 61

From the equation:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 62

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 63

Therefore: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 64

The above equation gives the linear charge density

Problem 5: potential charge

Using the poison’s partial differential equation:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 65

Where: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 66 is the Laplace operator while f and ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 67are real functions of the manifold

The Cartesian coordinates: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 68

Eliminating f, therefore: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 69

Using the Gauss law for electricity: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 70

Calculating for the electric field

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 71

Differentiating the equation: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 72

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 73

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 74

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 75

Thus the Electric charge is given as: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 76

The charge distribution: using the integral for the charge density: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 77 all over a line ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 78volume v, and surface s.

For line integral:ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 79

For surface integral:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 80

For volume integral

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 81

Calculating the homogenous charge density: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 82

The charge of any volume

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 83

So, ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 84

If the charge in the region has N discrete point-charges, therefore the charge distribution is given by:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 85

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 86

Thermal and Statistical Physics

Assignment #1

Question 1: quasi-static isothermal compression of a solid

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 87

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 88

But ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 89

Therefore: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 90ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 91

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 92 = 500atm

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 93 = ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 94atm = 0.15 ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 95

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 96ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 97

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 98ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 99 = ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 100

W = ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 101

Problem 2: work done on a gas

For an adiabatic process:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 102 i

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 103 ii

During the adiabatic process, P charges with V and the internal energy is given by:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 104 iii

Where: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 105 is the number of degrees of freedom divided by two and R is the universal gas constant.

Differentiating equation (iii) above and using the ideal gas law

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 106

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 107

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 108

Substituting (ii) and (iv) into equation (i):

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 109

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 110

Integrating the equation above:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 111

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 112

Thus: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 113

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 114 = constant

Work done, W =ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 115

Since the process is adiabatic: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 116

Therefore, from: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 117

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 118

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 119

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 120

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 121

Work done if the gas had leaked to the atmosphere

Question 3: work done on a gas

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 122

Volume of the cylinder: area ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 123

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 124

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 125

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 126

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 127

Question 4: isothermal expansion of a gas

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 128

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 129

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 130

The amount of heat supplied by the gas equals the amount of work done by the gas

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 131

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 132

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 133

Question 5: Quasi static expansion of an idea

For an adiabatic process:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 134 i

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 135 ii

During the adiabatic process, P charges with V and the internal energy is given by:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 136 iii

Where: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 137 is the number of degrees of freedom divided by two and R is the universal gas constant.

Differentiating equation (iii) above and using the ideal gas law

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 138

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 139

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 140

Substituting (ii) and (iv) into equation (i):

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 141

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 142

Integrating the equation above:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 143

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 144

Thus: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 145

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 146 = constant

Work done, W =ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 147

Since the process is adiabatic: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 148

Therefore, from: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 149

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 150

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 151

Replacing ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 152 with ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 153

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 154

Multiplying by (-) ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 155

Pf = ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 156

Pi = ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 157

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 158 = 3.16 ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 159

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 160

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 161

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 162

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 163368 Joules

Question 6: a gas engine cycle

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 164

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 165

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 166

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 167

Therefore the heat absorbed ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 168

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 169

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 170

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 171

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 172

Internal energy, ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 173RELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 174

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 175

U = 3566.706 Joules

Pressure of the gas after the temperature rise

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 176

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 177

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 178

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 179

For the adiabatic expansion:

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 180

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 181

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 182

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 183

Therefore the heat absorbed ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 184

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 185

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 186

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 187

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 188

Internal energy, ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 189RELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 190

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 191

U = 3566.706 Joules

Pressure ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 192of the gas after the temperature rise

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 193

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 194

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 195

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 196

Question 7: thermal efficiency

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 197

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 198

Therefore, thermal efficiency for a heat engine is the percentage of the heat energy that is transformed to work.

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 199

Using the second law of thermodynamics: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 200

Where: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 201 is the waste heat from the engine

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 202is the heat that enters the engine

At the end of the combustion process;

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 203

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 204ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 205

Therefore: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 206

Question 8:

Thermal efficiency

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 207

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 208

Therefore, thermal efficiency for a heat engine is the percentage of the heat energy that is transformed to work.

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 209

Using the second law of thermodynamics: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 210

Where: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 211 is the waste heat from the engine

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 212is the heat that enters the engine

At the end of the combustion process

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 213

Question 9: Thermal efficiency for diesel engine

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 214

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 215

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 216

All the work is done by the engine therefore giving a negative efficiency.

Question 10: Entropy

Change in entropy, ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 217

Where: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 218 Is the heat capacity given as 4180Jkg-1K‑1

Therefore: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 219

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 220

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 221

The entropy at the universe can be explained using the theory that at the state of the universe, it can be said to be the heat death of the universe.

Question 11: Maxwell’s equations

For equation M3

Using the faraday’s law ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 222

Differentiating the equation: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 223

Using the ampere’s law ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 224

Using the differential from: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 225

ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 226

Where: ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 227 and ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS 228

References

Ming, L. C., & Manghnani, M. H. (1978). Isothermal compression of bee transition metals

to 100kbar. J. Appl. Phys., 49, 207-212.

Zill, D. G., and Cullen, R. M. (2006). Advanced engineering mathematics. New York: Courier companies.