Math/Physic/Economic/Statistic Problems

 

Question 1: Electric field of a dipole Total: 25 marks
Two electric charges ๐‘ž1 and ๐‘ž2 of the opposite polarity are positioned along ๐‘ฅ-axis at ๐‘ฅ1 and ๐‘ฅ2 respectively.
a) Find analytically1 the location(s) (๐‘ฅ0,๐‘ฆ0) where the electric field ๐„ due to these charges is exactly zero2. Provide a line plot of ๐„(๐‘ฅ) which shows found location of ๐ธ =0 in relation to the charges. (8 marks)
b) Using MATLAB calculate numerically the distribution of the electric field ๐„(๐‘ฅ,๐‘ฆ) and electric potential ๐‘‰(๐‘ฅ,๐‘ฆ) in a rectangular domain3. Provide the plot of the unit vectors of the resulting vector field ๐„(๐‘ฅ,๐‘ฆ) overlaid onto
the filled colour contour plot ๐‘‰(๐‘ฅ,๐‘ฆ). Denote the location of the charges ๐‘ž1 and ๐‘ž2 on the plot with small coloured circles. Use reasonable resolution of the numerical grid, which provides clear representation of the electric field vectors as well as smooth contours of ๐‘‰. (15 marks)

c) Confirm the result of part (a) by clearly denoting the location(s) of (๐‘ฅ0,๐‘ฆ0) found in (a) on your plot from part (b) with a small marker. (2 marks)

Individual inputs for this question:

a) Take your Student ID number, e.g. SID=1234567
b) Add first 4 digits of SID together, keep adding until you end up with a single digit, i.e.
1+2+3+4=10=>1+0=1, assign this number to ๐ด
c) Add last 4 digits of SID together, keep adding until you end up with a single digit, assign it to ๐ต, i.e. ๐ต =4+5+6+7=22โ†’2+2=4. If you are ended up with ๐ด=๐ต, take ๐ต =๐ด+1.
d) Now find ๐ถ by dividing (๐ดโˆ’๐ต) by its modulus: ๐ถ =๐ดโˆ’๐ต/|๐ดโˆ’๐ต|, e.g. ๐ถ =1โˆ’4/|1โˆ’4|=โˆ’1;
e) Your values for Question 1 are: ๐‘ž1 =๐ถร—๐ตร—10โˆ’9 , ๐‘ž2 =โˆ’4๐‘ž1 Coulombs; ๐‘ฅ1 =โˆ’๐ด, ๐‘ฅ2 =๐ต metres.

1 computer algebra, such as symbolic mathematics toolbox of MATLAB, can be used if necessary. 2 besides the infinite distance away from the charges. 3 choose the size of the domain to include the location of (๐‘ฅ0,๐‘ฆ0) and cover some reasonable space around the charges: your plot should give a good presentation of the electric field vectors and contours of the potential.
This document is for Coventry University students for their own use in completing their assessed work for this module and should not be passed to third parties or posted on any website. Any infringements of this rule should be reported to facultyregistry.eec@coventry.ac.uk.

Question 2: Nodal analysis Total: 25 marks
In the circuit below (Fig.1), the electric potentials ๐‘‰1…๐‘‰10 at the corresponding nodes โ€œ1…10โ€ are measured with respect to the common node โ€œ1โ€. Assume the batteries have no internal resistance and their e.m.f. are ๐œ€1 =10 and ๐œ€2 =12 Volts.

To assign the values to resistors R, take first seven digits of your Student ID number, say ABCDEFG, and use the following values, in Ohms:
R1 = A + 3, R2 = B + 1, R3 = C + 4, R4 = D + 2, R5 = E + 2, R6 =F + 2, R7 = G + 1, R8 = A + B + 3, R9 = C+D+1, R10 = B+D+1, R11 = G+F+3, R12 = 2*R1+2, R13 = E+F+2.

 

Applying the Nodal Analysis and using the same nodesโ€™ numbers as in Fig.1:

a) use Kirchhoffโ€™s 1st rule and write the system of nodal equations for this circuit. Show assumed directions of the currents through batteries ๐œ€1 and ๐œ€2 (respectively ๐ผ14 and ๐ผ12) on the circuit (Fig.1). (5 marks)
b) Represent the problem in matrix form: show the system of equations in matrix form, including vector of unknowns, matrix of coefficients, vector of RHS. (5 marks)

Using MATLAB:

c) Solve the system of equations (b) using a matrix method of your choice, hence find the values of all nodal potentials ๐‘‰1…๐‘‰10 and the currents ๐ผ12 and ๐ผ14 through the batteries. (5 marks)
d) Find all individual currents through each of 15 elements of the circuit, denoting currents ๐ผ1…๐ผ13 through resistors ๐‘…1…๐‘…13 respectively. Show the correct directions of these currents on the sketch of the circuit with arrows. (5 marks)
e) Show that for the currents found in (d), the 1st Kirchhoffโ€™s rule is satisfied at each node of the circuit. (5 marks)

This document is for Coventry University students for their own use in completing their assessed work for this module and should not be passed to third parties or posted on any website. Any infringements of this rule should be reported to facultyregistry.eec@coventry.ac.uk.

Question 3: Motion of an electron in a magnetron Total: 25 marks
Consider a cross-section of two conducting coaxial cylinders in vacuum centred at the origin (Fig.2). The electric potential difference ฮ”๐‘‰ = ๐‘‰๐‘Ž โˆ’ ๐‘‰๐‘ is applied between them: the inner cylinder (the cathode) of radius ๐‘ has potential ๐‘‰๐‘ and the outer cylinder (the anode) of radius ๐‘Ž has potential ๐‘‰๐‘Ž. External magnetic field ๐, parallel to the axis of both cylinders is applied everywhere in space. From the surface of the cathode free electrons with charge โˆ’๐‘’ (where ๐‘’ > 0), and negligible initial kinetic energy are continuously emitted.
a) Starting from Laplaceโ€™s equation in polar coordinates, find the distribution of the electric potential ๐‘‰(๐‘Ÿ,๐œƒ) in the gap between the cylinders analytically (neglect the electric field of free electrons) and plot it with respect to the relevant coordinate(s) for ๐‘‰๐‘Ž =2000V, ๐‘‰๐‘ =โˆ’200 V, ๐‘ =0.02 m, ๐‘Ž=0.2m. (5 marks)
b) Assume the electron is at the distance ๐‘Ÿ from the centre (๐‘ โ‰ค ๐‘Ÿโ‰ค ๐‘Ž), has velocity vector ๐ฎ. Using your result from part (a), find the expression(s) for the full electromagnetic force acting on the electron of mass ๐‘š๐‘’ and charge โˆ’๐‘’. (5 marks)
c) The electric current through such a device is the amount of charge transferred from the cathode to the anode per unit of time. For a given non-zero voltage ฮ”๐‘‰, there is a certain threshold value of the magnetic field ๐ต =๐ต๐‘š๐‘Ž๐‘ฅ (โ€œHull cut-offโ€ value, in Tesla), above which the current stops flowing: ๐ต๐‘š๐‘Ž๐‘ฅ =โˆš8 ฮ”๐‘‰ ๐‘š๐‘’/๐‘’ ๐‘Ž(1โˆ’๐‘2 ๐‘Ž2).
Explain why such a cut-off effect is happening? (5 marks)
d) Write MATLABยฎ code, which simulates numerically the motion of a single electron emitted from the cathode and plot its trajectory between the electrodes. Use given in (a) values for constants ๐‘‰๐‘Ž, ๐‘‰๐‘, ๐‘ and ๐‘Ž.

i. Provide the code. (7 marks)
ii. Provide three plots of the electronโ€™s trajectory corresponding to three values of magnetic field ๐ต =ย  0.5๐ต๐‘š๐‘Ž๐‘ฅ,1.02๐ต๐‘š๐‘Ž๐‘ฅ,3๐ต๐‘š๐‘Ž๐‘ฅ (e.g. below, slightly above and way above the threshold ๐ต๐‘š๐‘Ž๐‘ฅ). Your results must
demonstrate the cut-off effect described above. Electrodes must be visible on the plots and the shape of the trajectories
must not be distorted. (3 marks)

This document is for Coventry University students for their own use in completing their assessed work for this module and should not be passed to third parties or posted on any website. Any infringements of this rule should be reported to facultyregistry.eec@coventry.ac.uk.

Question 4: Magnetic field due to a spherical coil Total: 25 marks

Besides a very long solenoid, there are few designs of magnetic induction coils aimed to produce more or less homogeneous magnetic field inside them: e.g. a pair of โ€œHelmholtz coilsโ€ or a set of three โ€œMaxwell coilsโ€. But what about a coil in a shape of a sphere? Letโ€™s take a sphere of radius ๐‘… =0.2 m centred at (0,0,0) with its axis being ๐‘ง-axis, so that the poles of the sphere are at ๐‘ง=ยฑ๐‘… (Fig.3). Now, beginning from the top pole (๐‘ง=๐‘…) we will wrap a single layer of ๐‘ =51 turns of thin wire with constant pitch4 along ๐‘‚๐‘ง around the sphere in the azimuthal direction (i.e. putting turns counter clockwise along the latitude of the sphere) and finish wrapping at the lower (โ€œsouthโ€) pole (๐‘ง=โˆ’๐‘…) so that
the surface of the sphere is uniformly covered with ๐‘ turns of the wire. Then we connect the ends of the wire to a source of the current ๐‘–, hence the total electric current in the coil is ๐ผ =๐‘–๐‘ [Ampere-turns] is flowing counter-clockwise
(shown in Fig.3 with the arrow). Alternatively, instead of wrapping a wire, you can imagine that the sphere has a thin conductive surface which carries almost purely azimuthal uniform current ๐ผ. It is suggested that the magnetic field inside such a โ€œspherical coilโ€ is constant and uniform, like the field inside a very long solenoid. Your objective is to provide an evidence that this suggestion is correct or wrong. You can assume, for simplicity, that each of ๐‘ turns of a wire in the coil isย  separate circular loop in ๐‘‚๐‘ฅ๐‘ฆ plane5 carrying the current ๐‘–. Complete the following tasks:

a) Find analytically6 the distribution of magnetic field ๐(๐‘ง) along the axis of the sphere ๐‘‚๐‘ง, both inside and outside of it, i.e. for |๐‘ง|<๐‘… and for |๐‘ง|โ‰ฅ ๐‘…. The result should be the line plot of ๐ต(๐‘ง) for e.g. โˆ’5๐‘…โ‰ค๐‘งโ‰ค5๐‘… and ๐ผ =103Ampere-turns. You can assume here that ๐‘ โ†’โˆž (i.e. the wire is very thin and is fully covering the surface of the sphere). (8 marks)

b) Using MATLAB, write a code, which utilising Biot-Savart law calculates numerically the magnetic field vectors in ๐‘‚๐‘ฅ๐‘ง plane both inside and outside such a spherical coil, e.g. on a rectangular grid โˆ’3๐‘…โ‰ค๐‘ฅ,๐‘งโ‰ค3๐‘….
Take the total current to be ๐ผ =10๐‘ [Ampere-turns]. (8 marks)

c) From your results of part (b) provide the plot of ๐(๐‘ฅ,๐‘ง) showing vectors of ๐ over the contours of |๐| on the same plot. (4 marks)
d) Produce the line plot of ๐(๐‘ง) distributions along the axis of the sphere, similar to the one from part (a), but using your numerical results from (b). While keeping the same value of the total current ๐ผ =103 [Ampere-turns], show several results of ๐(๐‘ง) for different number of turns ๐‘, e.g. ๐‘ = 5,10,50,100 on the same line plot. Compare with analytical solution from (a) by also plotting it in the same plot. Comment on your results.