Multivariable Module I: Analytical Geometry and the Vector

Matthew F. Causley, Ruben Hayrepetyan, Allan Taylor Kettering University

Contents

1 Introduction 2

2 Example: Stable Points in Space between the Sun and Earth2 2.1 Further investigation: Visualizing the Force ...... 3

3 Project: Finding the Other Points4

4 Example: The Hall Effect 5 4.1 the Hall Voltage...... 6 4.2 Hall Voltage Sensor: a Sample Calculation...... 6

5 Project: 7 5.1 The Biot Savart Law...... 7

1 1 Introduction

In this module we will use analytical geometry to formulate and solve problems arising in physics and engineering. This requires familiarity with vectors, to measure the direction and magnitudes of fundamental quantities such as distances, velocities and forces. When we measure the rates of change with respect to directional coordinates, the need for a gradient vector naturally arises. Below we will consider a problem in which the gradient vector is used to define critical points in orbital trajectories of satellites. This problem is not only very interesting, but also can be studied without complex calculations. By introducing basic notation along the way, we are able to focus on connecting mathematical operations and physical concepts.

2 Example: Stable Points in Space between the Sun and Earth

To conserve fuel, space satellites are often positioned at locations where the gravitational acceleration due to large masses (such as the sun and the earth, see Figure1) is minimal. There exists a point between the two bodies where the accelerations are equal and opposite (commonly called the Lagrangian point L1). We can compute this position in space using calculus.

Figure 1: Lagrange Points in the earth-sun plane. The point L1 lies on the line between the two bodies, and indicates where the gravitational accelerations balance out. In the right figure, potential lines are included, indicating the magnitude of gravitational forces. Source: Wikipedia

Consider the plane containing both the sun and the earth, with the sun centered at the point S(0, 0, 0), and the earth centered at E(xe, 0, 0). We will examine a satellite with mass m, located at a position (x, y, z). According to Newton’s law of gravitation, the forces acting on this point due to the sun and earth respectively, are

GMsm GMem Fs = 2 ˆr, Fe = 2 ˆre, (1) r re where G is the gravitational constant, Ms and Me are the masses of the sun and earth, respectively. The vectors and lengths are

hx, y, zi hxe − x, y, zi p 2 2 2 p 2 2 2 ˆr = , ˆre = , r = x + y + z , re = (xe − x) + y + z . (2) r re

Since the point L1 is collinear with the sun and the earth, we can assume it is at (x, 0, 0). Then, ˆr = ˆı, ˆre = −ˆı, and the force becomes a scalar quantity. If we also incorporate centrifugal forces, then the total force at L1 is   Ms Me Msx F = Gm − 2 + 2 + 3 . (3) x (xe − x) (xe)

2 Setting the force equal to zero produces a 5th order for x,

 2  3! GMsm Me x x 2 −1 + + = 0. (4) x Ms xe − x xe

But we are only interested in the one satisfying 0 < x < xe. Furthermore, we can reduce the difficulty of the problem by approximating the solution. We observe that two of the quantities are small, namely, x − x x M δ = e = 1 − ,  = e . xe xe Ms Then, the terms in parentheses from equation (4) can be written as

1 − δ 2 −1 +  + (1 − δ)3 = 0 δ  (1 − δ)2 + δ2 (1 − δ)3 = δ2  + δ2 (1 − 3δ) ≈ δ2  ≈ 3δ3.

Thus, we find that δ ≈ p3 /3, and in terms of the original variables we have r ! 3 Me x = xe 1 − . (5) 3Ms

2.1 Further investigation: Visualizing the Force Field In order to find the Lagrange point (5), we assumed that the satellite was located at (x, 0, 0), i.e. in the same plane of motion as the sun and the earth. What if we allow the y and z coordinates to vary? Then, the forces will act in different directions. We define the gravitational potential energy functions for the sun and earth, respectively as GMs GMe Vs(x, y, z) = − ,Ve(re) = − , (6) r re again using the definitions from equation (2). The force acting on the satellite will now be given by a vector F =< Fx,Fy,Fz >, which is given by taking of the potential functions with respect to each ∂ ∂ ∂ F = m (V + V ) ,F = m (V + V ) ,F = m (V + V ) . x ∂x s e y ∂y s e z ∂z s e rather than writing partial derivatives, we can use the gradient vector to define the force vector

 ∂ ∂ ∂  F = m∇V (x, y, z), ∇ = , , . ∂x ∂y ∂z

Upon evaluating the partial derivatives, we recover precisely (1)! For each point (x, y, z) in space, a force given by F(x, y, z) will act on the satellite. How do we visualize this? The Matlab script at the end of this module was used to produce Figure2, which shows the force field (1), with appropriately scaled coefficients. We can visualize the satellite as a marble, placed on the red plane in the Figure. If the marble is too close to either the sun or the earth, the marble will roll into them (retrograde), along paths determined by the gradient. Only at Lagrange points, such as L1, will the marble not roll (where the gradient vanishes). Here, the motion is stable, and so we sometimes call L1 a ”stable point”.

3 Figure 2: A plot of the force field (1). The satellite will move along the valleys, in the direction of .

1 %Compute and plot the gravitational potential of the earth and sun 2 %Written by Allan Taylor 3 %Kettering University 4 clc; clear; 5 6 %I modified the numbers to make the plot come out nice... 7 Me = 1;%5.972e24*5e6;%mass of earth in kg 8 Ms = 10;%1.989e30*1e2;%mass of sun in kg 9 G = 0.1;%6.673e−11;%gravitational constant inmˆ3/(kg *sˆ2) 10 xe = 2;%1.496e11;%distance from earth to sun inm 11 12 gridx = [−2:0.1:4];%[−2e11:1e10:+4e11]; 13 gridy = [−2:0.1:2];[−2e11:1e10:+2e11]; 14 [x,y] = meshgrid(gridx,gridy); 15 16 %compute distances 17 re = sqrt((x−xe).ˆ2 + (y−0).ˆ2); 18 rs = sqrt((x−0).ˆ2 + (y−0).ˆ2); 19 20 %acceleration of gravity at that point 21 ge = −G*Me./(re.ˆ2); 22 gs = −G*Ms./(rs.ˆ2); 23 24 % cap acceleration of gravity to −10m/secˆ2 25 ge(ge<−10) = −10;% Magic 26 gs(gs<−10) = −10;% Magic 27 28 %Plot the data 29 figure(1); 30 mesh(gridx,gridy,ge+gs) 31 axis tight;

3 Project: Finding the Other Points

In order to find the other 4 Lagrange points, the full force balance equation must be investigated. Your project will involve the following tasks.

1. Beginning with equation (6), compute the forces Fs = ∇Vs(x, y, z), and Fe = ∇Ve(x, y, z).

2. The sun and the earth are attracted to each other by a force equal to Fs(xe, 0, 0). Compute this force.

4 2 Meve 3. The centrifugal force of the earth repels it from the sun. This force is of the form Fc(x, y, z) = 2 ˆre. xe Compute this force.

4. Because the earth is in a stable orbit, the forces are balanced. Set Fs(xe, 0, 0) = Fc(xe, 0, 0), and solve for the velocity ve of the earth.

5. Compute the orbital period of the earth, T = 2πxe . e ve 6. Now consider a satellite of mass m, located at a Lagrange point L(x, y, z). Both the sun and earth will mv2 exert attractive forces on this point. Furthermore, it will also have a centrifugal force F (x, y, z) = ˆr. cs r2 The force balance is Fs(x, y, z) = Fe(x, y, z) + Fcs(x, y, z). Substitute your expressions, and obtain the force balance. Use this to solve for the satellite velocity v.

2πr 7. Compute the orbital period of the satellite, T = v . 8. In order for the point L(x, y, z) to be a Lagrange point, the orbital period must be the same as that of earth. Set T = Te, and establish a relationship between v and ve.

9. Returning to the force balance for the satellite, eliminate ve in favor of v, and then eliminate v using the force balance between the earth and the sun. You should now have an expression involving only x, y, z, the masses of the sun and earth, and the distance xe between them. 10. Obtain approximate solutions (x, y, z), using the procedure resulting in equation (5). Note that this is one of 5 possible solutions, and one of 3 for which y = z = 0.

Examples and Applications Look up the relevant values, such as the distance between the earth (or other planets) and the sun, along with their masses. Discuss numerical examples, and describe possible applications of your results.

References Include all sources you used. For example:

1. ”Lagrangian point.” Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. 22 July 2004. Web. 10 Nov. 2016.

2. Koon, W. S.; Lo, M. W.; Marsden, J. E.; Ross, S. D. (2006). Dynamical Systems, the Three-Body Problem, and Space Mission Design. p. 9.

4 Example: The Hall Effect

In many electronic systems, it is useful for us to know the electrical current flowing through a particular part of the circuit. Current measurements may be used for control purposes (e.g. in electric motors, or solenoids) or they can be used to protect a particular circuit; if a large current is detected, a safety routine can be employed. One popular way of measuring current through a conductor is based on the Hall Effect. When a current-carrying conductor is placed in a magnetic field, a transverse force is exerted on the moving charges, causing them to crowd on one side of the conductor. Referring to Figure4, consider a conductor of dimensions `, w, and t, through which a current I (measured in Amps) is flowing. A magnetic field B (in Tesla) is applied, and the resulting magnetic force F (in Newtons) acts on the charges. The magnetic force is given by a cross product F = ` (I × B), that is

ˆı ˆ kˆ

hFx,Fy,Fzi = ` Ix Iy Iz

Bx By Bz

5 With the orientation shown in Figure4, we have I = hIx, 0, 0i, B = h0,By, 0i, and so F = h0, 0,Fzi. That is, the force Fz = `IxBy is in the z-direction, and so the positive and negative charges will crowd towards the top and bottom of the conductor.

4.1 the Hall Voltage

The separation of charge results in an electric field in the z-direction and thus a voltage-difference VH (the Hall voltage) between the top and bottom ends of the conductor. The forces due to electric and magnetic fields on a particle with charge q are expressed by the Lorentz Force equation

F = qE + qv × B . (7) |{z} | {z } Fe Fm Here E, is the electric field intensity (in Volts per meter), and v is the charge velocity (in meters per second). Inside a conductor, the average velocity of charges, or drift velocity vd is given by I v = d neA where n is the charge carrier density of the conductor (in electrons per cubic meter), e is the fundamental charge of an electron (in Coulombs), and A = wt is the cross-sectional of the conductor (in square meters). As 22 an example, the carrier density of copper is 8.42 × 10 electrons per cubic meter. Setting v = vd, we find the magnetic force q F = I × B m neA After some time, eventually the charge distribution in the metal will stabilize. The net force (7) on the charges will equal zero, and so Fe = −Fm. (8)

Finally, the Hall voltage is determined by the electric field VH = E · x, where x is a displacement vector. From the configuration shown in Figure4, we obtain VH = Ezw, and we solve for this quantity from the force balance equation qI B I B qE = − x y =⇒ V = − x y . (9) z newt H net

4.2 Hall Voltage Sensor: a Sample Calculation Consider the case of a Hall Voltage sensor consisting of a piece of copper foil, placed within a static magnetic field similar to the geometric arrangement shown in Figure4. Assume the current to be Ix = 1 Ampere, the thickness of the copper foil to be t = 100µm (the thickness of a human hair), and the magnetic flux density to be By = 1 Tesla. The resulting Hall Voltage would be 1A · 1T V = − = −1.2 × 10−5V. H 8.42 × 1022m−3 · 1.6 × 10−19C · 100 × 10−6m

6 This is an extremely small voltage, which can be very difficult to measure without the right equipment. Usually, a thin piece of doped silicon is used as a conductor rather than a metal. This can drastically reduce the carrier concentration, which will increase the Hall Voltages to more easily measurable levels (i.e. in the millivolt range). Why is a Hall Voltage Sensor so useful? The answer lies in equation (9): The Hall Voltage is directly proportional to the conducted current! Thus, if we want to measure the current in a circuit somewhere, we can simply pass it through this sensor, and measure the Hall Voltage.

5 Project:

Definitions In this project, you will need to review some basic definitions from your Electricity and Magnetism course relating to forces on charged particles. You can use books or internet resources such as Wikipedia.

5.1 The Biot Savart Law In the first section of your project, define and explain the concepts of the Lorentz Force Law in your own words. Comment on both terms: the electric force term and the magnetic force term. Consider a Hall Effect sensor made with a piece of Silicon which has been doped to a carrier concentration of 3 × 1015 electrons per cubic meter. The thickness is 25µm and the current is 1 mA. Answer the following questions, which relate to the geometrical configuration of the system.

1. What will be the Hall Voltage for the system described above if the magnetic flux density passing through the Silicon is 1 T? 2. Now, assume that the orientation of this magnetic field is rotated by 45 deg in the xy-plane. Now what will the Hall Voltage be? 3. If the current flows in the vector direction h5, 0, 1i and the magnetic field has a vector direction h−1, 6, −2i, what will be the direction of the applied magnetic force?

4. A circular coil of wire with a radius of a meters, a current of Ic Amperes, and N number of turns (see Figure 5.1) will produce a magnetic flux density of B Tesla equal to I N B = µ c , 0 2a −7 at the center of the circular coil, where µ0 = 4π × 10 is the permeability of free space (in Henrys per meter). Assume you place the silicon conductor at the center of the coil and construct the coil with a radius of 100mm and 2000 turns of wire. What will be VH /Ic, the ratio between the Hall Voltage and the coil current?

7 Examples and Applications Discuss numerical examples. Describe possible applications of your results.

References Include all sources you used. For example:

1. H. Zumbahlen, Analog Devices Inc., Linear Circuit Design Handbook, Elsevier Science, 2008, ISBN: 9780750687034.

2. S. Ball, C. Huddleston, D. Ibrahim, E. Ramsden, & J. S. Wilson, Chapter 1. Hall-Effect Physics. In, Test and Measurement: Know It All (pp. 1-10). Elsevier Science and Technology, 2008, ISBN: 978-1856175302.

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