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# 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

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)

1. 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

1. 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 point-charges, therefore the charge distribution is given by:

Thermal and Statistical Physics

Assignment #1

Question 1: quasi-static isothermal compression of a solid

But

Therefore:

= 500atm

= atm = 0.15

=

W =

Problem 2: work done on a gas

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 =

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

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 =

Therefore, from:

Replacing with

Multiplying by (-)

Pf =

Pi =

= 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

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-1K‑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

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, 207-212.

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