ELCTROMAGNETIC THEORY AND PLASMA PHYSICS & THERMAL AND STATISTICAL PHYSICS Essay Example
 Category:Physics
 Document type:Math Problem
 Level:Masters
 Page:2
 Words:910
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

For spherical the angle the zaxis 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 xy plane and the xaxis is φ.
Therefore for the function in spherical coordinate system:
The spherical coordinates (r, θ,φ)

For a cylindrical coordinate system:
Cylindrical coordinates (ρ, φ, z)
Problem 2
Since there is full spherical symmetry, the derivatives with respect to ? and ϕ must be zero
Total charge is Q = _{}
Using the Laplace law: _{}
Solution of form: _{}
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.
Using the form
Substituting in the poison’s equation it gives:
Giving _{}
Using boundary conditions at the surface of the sphere: r = R
Thus giving: _{}
The potential inside the sphere is given as:
Thus the total charge is _{}qsin_{}
Problem 3

Evaluate

_{}
_{} = 0
_{}
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:
This relation explains how the charge density varies in many dimensions.
For a homogenous charge density:
From the equation:
Therefore: _{}
The above equation gives the linear charge density
Problem 5: potential charge
Using the poison’s partial differential equation:
Where: _{} is the Laplace operator while f and _{}are real functions of the manifold
The Cartesian coordinates: _{}
Eliminating f, therefore: _{}
Using the Gauss law for electricity: _{}
Calculating for the electric field
Differentiating the equation: _{}
Thus the Electric charge is given as: _{}
The charge distribution: using the integral for the charge density: _{} all over a line _{}volume v, and surface s.
For line integral:
For surface integral:
For volume integral
Calculating the homogenous charge density: _{}
The charge of any volume
So, _{}
If the charge in the region has N discrete pointcharges, therefore the charge distribution is given by:
Thermal and Statistical Physics
Assignment #1
Question 1: quasistatic isothermal compression of a solid
But _{}
Therefore: _{}_{}
_{} = 500atm
_{} = _{}atm = 0.15 _{}
_{}_{}
_{}_{} = _{}
W = _{}
Problem 2: work done on a gas
For an adiabatic process:
_{} i
_{} ii
During the adiabatic process, P charges with V and the internal energy is given by:
_{} iii
Where: _{} 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
Substituting (ii) and (iv) into equation (i):
Integrating the equation above:
Thus: _{}
_{} = constant
Work done, W =_{}
Since the process is adiabatic: _{}
Therefore, from: _{}
_{}
Work done if the gas had leaked to the atmosphere
Question 3: work done on a gas
Volume of the cylinder: area _{}
Question 4: isothermal expansion of a gas
The amount of heat supplied by the gas equals the amount of work done by the gas
Question 5: Quasi static expansion of an idea
For an adiabatic process:
_{} i
_{} ii
During the adiabatic process, P charges with V and the internal energy is given by:
_{} iii
Where: _{} 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
Substituting (ii) and (iv) into equation (i):
Integrating the equation above:
Thus: _{}
_{} = constant
Work done, W =_{}
Since the process is adiabatic: _{}
Therefore, from: _{}
Replacing _{} with _{}
Multiplying by () _{}
P_{f } = _{}
P_{i} = _{}
_{} = 3.16 _{}
_{}368 Joules
Question 6: a gas engine cycle
Therefore the heat absorbed _{}
Internal energy, _{}R_{}
_{}
U = 3566.706 Joules
Pressure of the gas after the temperature rise
For the adiabatic expansion:
Therefore the heat absorbed _{}
Internal energy, _{}R_{}
_{}
U = 3566.706 Joules
Pressure _{}of the gas after the temperature rise
Question 7: thermal efficiency
Therefore, thermal efficiency for a heat engine is the percentage of the heat energy that is transformed to work.
Using the second law of thermodynamics: _{}
Where: _{} is the waste heat from the engine
_{}is the heat that enters the engine
At the end of the combustion process;
_{}_{}
Therefore: _{}
Question 8:
Thermal efficiency
Therefore, thermal efficiency for a heat engine is the percentage of the heat energy that is transformed to work.
Using the second law of thermodynamics: _{}
Where: _{} is the waste heat from the engine
_{}is the heat that enters the engine
At the end of the combustion process
Question 9: Thermal efficiency for diesel engine
All the work is done by the engine therefore giving a negative efficiency.
Question 10: Entropy
Change in entropy, _{}
Where: _{} Is the heat capacity given as 4180Jkg^{1}K^{‑1}
Therefore: _{}
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 _{}
Differentiating the equation: _{}
Using the ampere’s law _{}
Using the differential from: _{}
Where: _{} and _{}
References
Ming, L. C., & Manghnani, M. H. (1978). Isothermal compression of bee transition metals
to 100kbar. J. Appl. Phys., 49, 207212.
Zill, D. G., and Cullen, R. M. (2006). Advanced engineering mathematics. New York: Courier companies.