matlab

Setting Tutors Abdeldjalil Bennecer 18/11/2016

Moderating Tutors Phil Picton 18/11/2016

Department of Engineering & Technology

School of Science & Technology - ASSIGNMENT BRIEF -

Module Code: ENG2054

Module Name: Electromagnetism for Engineers

Assignment No: Assessment 1 Weighting: 50%

Submission Date/Time:

18/12/2016 23:59

Learning Outcomes:

Please refer to Module Specifications at: http://oldweb.northampton.ac.uk/caf/Undergraduate%20

Module%20Specifications/Engineering%20Manufacture%2 0&%20Quality/

Assignment:

Objectives:

1. Use Matlab to evaluate the effect of electric dipole charge Q and distance d on the electric field intensity and potential.

2. Use Matlab to plot the electric field and equipotential lines due to:

a. Two point charges Q and -3Q located at (x, y) = (-2, 0) and (1, 0) respectively

b. Four point charges Q, -Q, Q, -Q located at (x, y) = (-1, -1), (1, -1), (1, 1), (-1, 1)

For simplicity, assume for the second objective that Q = 4πε0 and -10 < x,

y < 10 Theory:

Electric dipole is two point charges of equal magnitude and opposite polarity

separated by a very small distance compared to the distance where the field is calculated.

Setting Tutors Abdeldjalil Bennecer 18/11/2016

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The electric far field denoted in spherical coordinate system (r, θ, φ)

is given by: �⃗� = 𝑄𝑑

4𝜋𝜀0𝑟 3 (2 cos 𝜃 𝑒�̂� + sin 𝜃 𝑒�̂�)

In Cartesian coordinates, where the dipole axis is along z,

�⃗� = 𝑄𝑑

4𝜋𝜀0𝑟 3 (3 cos 𝜃 sin 𝜃 𝑖̂ + 3 cos 𝜃 sin 𝜃 𝑗̂ + [3 cos2 𝜃 − 1]�̂�)

The electric potential due to a dipole is: 𝑉 = 𝑄𝑑 cos 𝜃

4𝜋𝜀0𝑟 2

The most commonly used numerical methods in electromagnetism are moment

method (to solve integral equations), finite element method and finite distance method (to solve partial difference equations). Despite the approximate nature of the numerical methods, they are sufficiently accurate for engineering

problems.

For N co-planar point sources located at position vectors r1, r2, r3, …,rN, the electric field and electric potential are given respectively by:

Setting Tutors Abdeldjalil Bennecer 18/11/2016

Moderating Tutors Phil Picton 18/11/2016

𝐸 = ∑ 𝑄𝐾

4𝜋𝜀0

𝑟 − 𝑟𝐾⃗⃗ ⃗

|𝑟 − 𝑟𝐾⃗⃗ ⃗| 3

𝑁

𝐾=1

And

𝑉 = ∑ 𝑄𝐾

4𝜋𝜀0|𝑟 − 𝑟𝐾⃗⃗ ⃗|

𝑁

𝐾=1

If the charges are on the same plane (z is constant), the above equations are simplified to

𝐸 = ∑

𝑄𝐾 4𝜋𝜀0

[(𝑥 − 𝑥𝐾)𝑖̂ + (𝑦 − 𝑦𝐾)𝑗̂]

[(𝑥 − 𝑥𝐾) 2 + (𝑦 − 𝑦𝐾)

2] 3

2⁄

𝑁

𝐾=1

(1)

And

𝑉 = ∑

𝑄𝐾

4𝜋𝜀0[(𝑥 − 𝑥𝐾) 2 + (𝑦 − 𝑦𝐾)

2] 1

2⁄

𝑁

𝐾=1

(2)

Procedure:

Use permutations of following ranges to calculate the effects of varying the

values of charge and distance: Q = [1.5, 2.5, 5, 10, 20] μC

d = [0.25, 0.5, 1, 2, 3, 4] mm

Choose a range of x, y, and z such that:

 To avoid singularity points i.e. the electric field or potential are undefined (e.g. tends to infinity)

 The points generated are not too close compared to dipole distance

It may be easier to convert (x, y, z) to spherical coordinates in order to work out the electric field.

In order to plot the electric field lines, follow these steps:

1. Choose a starting point

2. Calculate Ex and Ey at that point using equation (1)

3. Move along the field line from the old point (x, y) to a new point (x’, y’) =

Setting Tutors Abdeldjalil Bennecer 18/11/2016

Moderating Tutors Phil Picton 18/11/2016

(x+Δx, y+Δy). The movements Δx and Δy along x and y are given by:

∆𝑥 = 𝐸𝑥∆𝑙

[𝐸𝑥 2 + 𝐸𝑦

2] 1

2⁄

∆𝑦 = 𝐸𝑦∆𝑙

[𝐸𝑥 2 + 𝐸𝑦

2] 1

2⁄

The value of incremental length Δl is crucial for accurate plots. Although the

smaller the value, the more accurate are the plots, it should be noted that it may cause a heavy computational load and require considerable storage. A value approximately equal to 0.1 should yield adequate results.

4. Repeat the calculations in step 2 and continue to generate new points until a line is completed within a given (x, y) range

5. Once the line is complete repeat step 1 by choosing another starting point. Note that there an infinite number of infinite lines and therefore any point is likely to be on a field line.

In order to plot the potential lines, follow these steps:

1. Choose a starting point

2. Calculate Ex and Ey components of E at that point using equation (1)

3. Move along the equipotential line (perpendicular to the electric field lines) from the old point (x, y) to a new point (x’, y’) = (x+Δx, y+Δy). The

movements Δx and Δy along x and y are given by:

∆𝑥 = −𝐸𝑦∆𝑙

[𝐸𝑥 2 + 𝐸𝑦

2] 1

2⁄

∆𝑦 = −𝐸𝑥∆𝑙

[𝐸𝑥 2 + 𝐸𝑦

2] 1

2⁄

You can check the accuracy of the new point by calculating the potential at the new and old points using equation (2); they must be equal because they are on the same equipotential line.

4. Repeat the calculations in step 2 and continue to generate new points until a line is completed within a given (x, y) range

5. Once the line is complete repeat step 1 by choosing another starting point. Note that there an infinite number of infinite lines and therefore any point is likely to be on a field line.

Setting Tutors Abdeldjalil Bennecer 18/11/2016

Moderating Tutors Phil Picton 18/11/2016

Check that there are no singularity points and the points generated are not too close to the charges.

Deliverables:

Write a well-documented Matlab code which take a position point as an input, the values of Q and d and then plot properly labelled and titled graphs representing the electric field and electric potential in terms of Q and d.

Typical graphs for E and V would look like these in that order:

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1 0

0.5

1

1.5

2

x 10 7

Setting Tutors Abdeldjalil Bennecer 18/11/2016

Moderating Tutors Phil Picton 18/11/2016

Write a well-documented Matlab code that plots properly labelled and titled graphs representing the electric field lines and electric equipotential lines for the

two point charges and quadpole.

Write an analytical report about your observations on the effect of changing Q

and d on the electric field strength and electric potential including the values that yields the strongest field by inspection of the graph. Also include comments on the variations of the electric field intensity and voltage in arbitrary two points

charge and quadpole. You may experiment with other scenarios of different charge values and different locations.

Please present your answer in the form of a report. Additionally 10% of the

marks is given for organisation and presentation. You will be expected to show evidence of research and well balanced arguments supported by academic literature relevant to the problem (in addition to the class learning material).

Assessment Criteria:

Your assignment will be assessed in accordance with University Modular Framework (UMF) grading criteria, and will be awarded a letter grade. Specific criteria for this assignment showing what is required to achieve a grade in each of the grade categories are shown below: http://www.northampton.ac.uk/staff/academic-and-curriculum-administration/exams-and-assessment/generic-grade-criteria

Grade Specific Criteria

A A very strong distinction: Work of distinguished quality, correct technically, well

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1 -1.5

-1

-0.5

0

0.5

1

1.5

x 10 5

Setting Tutors Abdeldjalil Bennecer 18/11/2016

Moderating Tutors Phil Picton 18/11/2016

referenced and visually distinctive. B A very strong merit: Work of commendable quality based on a strong

factual/conceptual knowledge and understanding of the concepts and related theory.

C A merit: Work of sound quality based on a firm factual and conceptual understanding of techniques and related theory. The work may be rather standard, but will be mostly accurate both written and figures presented.

D A pass: Work of broadly satisfactory quality covering adequately the factual and/or conceptual understanding of the concepts and related theory.

F A fail: Work that falls well short of the threshold standards in relation to one or more area of knowledge

G Nothing presented or less than 4%

Intermediate grades also apply i.e. A+, A, A-, B+, B, B- etc Numerically marked assignments are converted to a letter grade using the following:

Score % Letter Grade Score % Letter Grade

80-100 A+ 50-52 C-

75-79 A 47-49 D+

70-74 A- 43-46 D

67-69 B+ 40-42 D-

63-66 B 35-39 F+

60-62 B- 20-34 F

57-59 C+ 5-19 F-

53-56 C 0-4 G