Write MATLAB® 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 𝑉𝑎, 𝑉𝑐, 𝑐 and 𝑎.

Question 1: Electric field of a dipole Total: 25 marks

Two electric charges 𝑞1 and 𝑞2 of the opposite polarity are positioned along 𝑥–axis at 𝑥1 and 𝑥2 respectively.

a) Find analytically1 the location(s) (𝑥0,𝑦0) where the electric field 𝐄 due to these charges is exactly zero2. Provide a line plot of 𝐄(𝑥) which shows found location of 𝐸 =0 in relation to the charges. (8 marks)

b) Using MATLAB calculate numerically the distribution of the electric field 𝐄(𝑥,𝑦) and electric potential 𝑉(𝑥,𝑦) in a rectangular domain3. Provide the plot of the unit vectors of the resulting vector field 𝐄(𝑥,𝑦) overlaid onto the filled colour contour plot 𝑉(𝑥,𝑦).

Denote the location of the charges 𝑞1 and 𝑞2 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 𝑉.
(15 marks)
Code:8 marks
Plot:7 marks

c) Confirm the result of part (a) by clearly denoting the location(s) of (𝑥0,𝑦0)
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 𝐴

c) Add last 4 digits of SID together, keep adding until you end up with a single digit, assign it to 𝐵, i.e. 𝐵 =4+5+6+7=22→2+2=4. If you are ended up with 𝐴=𝐵, take 𝐵 =𝐴+1.

d) Now find 𝐶 by dividing (𝐴−𝐵) by its modulus: 𝐶 =𝐴−𝐵/|𝐴−𝐵|, e.g. 𝐶 =1−4/|1−4|=
−1;

e) Your values for Question 1 are: 𝑞1 =𝐶×𝐵×10−9 , 𝑞2 =−4𝑞1 Coulombs; 𝑥1 =−𝐴, 𝑥2 =𝐵 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 (𝑥0,𝑦0) 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.

Question 2: Nodal analysis Total: 25 marks

In the circuit below (Fig.1), the electric potentials 𝑉1…𝑉10 at the corresponding nodes
“1…10” are measured with respect to the common node “1”. Assume the batteries have
no internal resistance and their e.m.f. are 𝜀1 =10 and 𝜀2 =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.
Fig.1

Applying the Nodal Analysis and using the same nodes’ numbers as in Fig.1:

a) use Kirchhoff’s 1st rule and write the system of nodal equations for this circuit. Show assumed directions of the currents through batteries 𝜀1 and 𝜀2 (respectively 𝐼14 and 𝐼12) 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 𝑉1...𝑉10 and the currents 𝐼12 and 𝐼14 through the batteries. (5 marks)

d) Find all individual currents through each of 15 elements of the circuit, denoting currents 𝐼1…𝐼13 through resistors 𝑅1…𝑅13 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 Kirchhoff’s rule is satisfied at each node of the circuit. (5 marks)

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 Δ𝑉 = 𝑉𝑎 − 𝑉𝑐 is applied between them: the inner cylinder (the cathode) of radius 𝑐 has potential 𝑉𝑐 and the outer cylinder (the anode) of radius 𝑎 has potential 𝑉𝑎. External magnetic field 𝐁, parallel to the axis of both cylinders is applied everywhere in space. From the surface of the cathode free electrons with charge −𝑒 (where 𝑒 > 0), and negligible initial kinetic energy are continuously emitted.

a) Starting from Laplace’s equation in polar coordinates, find the distribution
of the electric potential 𝑉(𝑟,𝜃) 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 𝑉𝑎 =2000V, 𝑉𝑐 =−200 V, 𝑐 =0.02 m, 𝑎=0.2m. (5 marks)

b) Assume the electron is at the distance 𝑟 from the centre (𝑐 ≤ 𝑟≤ 𝑎), has velocity vector 𝐮. Using your result from part (a), find the expression(s) for the full electromagnetic force acting on the electron of mass 𝑚𝑒 and charge −𝑒. (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 Δ𝑉, there is a certain threshold value of the magnetic field 𝐵 =𝐵𝑚𝑎𝑥 (“Hull cut–off” value, in Tesla), above which the current stops flowing: 𝐵𝑚𝑎𝑥 =√8 Δ𝑉 𝑚𝑒/𝑒
𝑎(1−𝑐2
𝑎2)
.
Explain why such a cut–off effect is happening? (5 marks)

d) Write MATLAB® 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 𝑉𝑎, 𝑉𝑐, 𝑐 and 𝑎.

i. Provide the code.
(7 marks)

ii. Provide three plots of the electron’s trajectory corresponding to three values of magnetic field 𝐵 =0.5𝐵𝑚𝑎𝑥,1.02𝐵𝑚𝑎𝑥,3𝐵𝑚𝑎𝑥 (e.g. below, slightly above and way above the threshold 𝐵𝑚𝑎𝑥). 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)

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 “Helmholtz coils” or a set of three “Maxwell coils”. But what about a coil in a shape of a sphere?

Let’s take a sphere of radius 𝑅 =0.2 m centred at (0,0,0) with its axis being 𝑧–axis, so that the poles of the sphere are at 𝑧=±𝑅 (Fig.3). Now, beginning from the top pole (𝑧=𝑅) we will wrap a single layer of 𝑁 =51 turns of thin wire with constant pitch4 along 𝑂𝑧 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 (“south”) pole (𝑧=−𝑅) so that the surface of the sphere is uniformly covered with 𝑁 turns of the wire.

Then we connect the ends of the wire to a source of the current 𝑖, hence the total electric current in the coil is 𝐼 =𝑖𝑁 [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 𝐼.

It is suggested that the magnetic field inside such a “spherical coil” 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 𝑁 turns of a wire in the coil is a separate circular loop in 𝑂𝑥𝑦 plane5 carrying the current 𝑖. Complete the following tasks:

a) Find analytically6 the distribution of magnetic field 𝐁(𝑧) along the axis of the sphere 𝑂𝑧, both inside and outside of it, i.e. for |𝑧|<𝑅 and for |𝑧|≥𝑅. The result should be the line plot of 𝐵(𝑧) for e.g. −5𝑅≤𝑧≤5𝑅 and 𝐼 =103Ampere–turns. You can assume here that 𝑁 →∞ (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 𝑂𝑥𝑧 plane both inside and outside such a spherical coil, e.g. on a rectangular grid −3𝑅≤𝑥,𝑧≤3𝑅. Take the total current to be 𝐼 =10𝑁 [Ampere–turns]. (8 marks)

c) From your results of part (b) provide the plot of 𝐁(𝑥,𝑧) showing vectors of 𝐁 over the contours of |𝐁| on the same plot. (4 marks)

d) Produce the line plot of 𝐁(𝑧) 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 𝐼 =103 [Ampere–turns], show several results of 𝐁(𝑧) for different number of turns 𝑁, e.g. 𝑁 =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.