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1 PH504: 1. Mathematics The mathematics required for PH504. Objective: to provide a revision of the most important and relevant points. It is not intended to teach the mathematics from scratch. For more detailed treatments consult your first year mathematics’ notes or textbooks. Some of the electromagnetism textbooks provide a chapter or appendix covering the required mathematics. 1. Complex Numbers. Postive Scalars. The voltage produced by a battery is a scalar quantity. So is the resistance of a piece of wire (ohms), or the current through it (amps). Negative numbers: debt We need square roots to evaluate the diagonal length of a square. The square root of a negative number? IMAGINERY or Imposssible Number However, when we begin to analyze alternating current circuits, we find that quantities of voltage, current, and even resistance (called impedance in AC) are not the familiar one-dimensional quantities we are used to measuring in DC circuits. Rather, these quantities, because they're dynamic (alternating in direction and amplitude), possess other dimensions that must be taken into account. Frequency and phase shift are two of these dimensions that come into play. Even with relatively simple AC circuits, where we are only dealing with a single frequency, we still have the dimension of phase shift to contend with in addition to the amplitude. 1 2 Complex numbers. Here is where we need to abandon scalar numbers for something better suited: complex numbers. Just like the example of giving directions from one city to another, AC quantities in a singlefrequency circuit have both amplitude (analogy: distance) and phase shift (analogy: direction). A complex number is a single mathematical quantity able to express these two dimensions of amplitude and phase shift at once Vectors. .When used to describe an AC quantity, the length of a vector represents the amplitude of the wave while the angle of a vector represents the phase angle of the wave relative to some other (reference) waveform. Complex numbers are useful for AC circuit analysis because they provide a convenient method of symbolically denoting phase shift between AC quantities like voltage and current. i or j = square root of –1. A complex number consists of a real and an imaginary part. Both the real and imaginary parts are real numbers, but the imaginary part is multiplied with the square root of -1. Complex numbers can be expressed in numerous forms. 1. Rectangular Form A complex number in rectangular form looks like this A = a + jb where a is the real part and b is the imaginary part. j is sqrt(-1). The concept is: Imaginery Numbers occupy an axis perpenicular to that occupied by Real Numbers. Adding and subtracting complex numbers in rectangular form is carried out by adding or subtracting the real parts and then adding and subtracting the imaginary parts. (5 + j2) + (2 - j7) = (5 + 2) + j(2 - 7) = 7 - j5 (2 + j4) - (5 + j2) = (2 - 5) + j(4 - 2) = -3 + j2 2 3 Multiplying is slightly harder than addition or subtraction. It must be carried out like the multiplication of two binomials, multiplying both parts of one by both parts of the other. When you multiply the imaginary parts, j*j = sqrt(-1) * sqrt(-1) = -1, so that part becomes real). (2 + j2) (8 - j3) = (2 * 8) + j(2 * 8) + j(2 * -3) + j*j (2 * -3) = 16 + j16 - j6 + 6 = 22 + j10 Division requires a new idea to be introduced. The complex conjugate of a number is the number that has the same real part as the original number but an imaginary part that differs only in its sign. The complex conjugate is denoted by an asterisk immediately following the number or variable. A = (2 + j2) A* = (2 + j2)* = 2 - j2 When dividing two complex numbers, you must first multiply both the numerator and denominator by the complex conjugate of the denominator. This multiplication results in a denominator that has only a real part. (4 + j3) / (2 + j2) = ((4 + j3) (2 - j2)) / ((2 + j2) (2 - j2)) = ((8 + 6) + j(8 - 6)) / ((4 + 4) + j(4 - 4)) = (14 + j2) / 8 = 7/4 + j/4 2. Polar Form Another way to represent complex numbers is in polar form. If you look at the real and imaginary parts of a complex number as coordinates in a plane, then the real part would be the x coordinate and the imaginary part the y coordinate. In rectangular form, the x and y coordinate are specified in that way. 3 4 In polar form, the point in the plane is instead defined by a magnitude and an angle. Polar form relates to rectangular form in the following way. (magnitude) r = sqrt(a² + b²) (angle) theta = tan-1 b/a and a = r cos (theta) b = r sin (theta) . A complex number is then represented as A = r | theta where r = magnitude = |A| and theta = angle = ang A Multiplication and division are much simpler for numbers in polar form To find the conjugate of a complex number in polar form, simply reverse the sign of the angle. A = 5 | 25° A* = 5 | -25° With multiplication you multiply the magnitudes and add the angles, and with division you divide the second magnitude from the first and subtract the second angle from the first. (5 | 45° ) (2 | 20°) = (5)(2) | (45 + 20)° = 10 | 65° (4 | 90°) / (2 | 45°) = 4/2 | (90 - 45)° = 2 | 45° To add or subtract we basically need to convert the numbers back into rectangular form 4 5 (2 | 45°) + (8 | 30°) = (2 cos 45° + 8 cos 30°) + j(2 sin 45° + 8 sin 30°) = 8.342 + j2.414 The answer can then be converted back to polar form if desired r = sqrt(8.342² + 2.414²) = 8.684 theta = tan-1 (2.414/8.342) = 16.1° (2 | 45°) + (8 | 30°) = 8.684 | 16.1° If possible it is best to add and subtract in rectangular form and multiply and divide in rectangular form. 3. Complex Exponential Form The complex exponential form is another way of representing a complex number. The following formula shows how it relates to rectangular and polar form e j = cos () + j sin () = 1 | The complex exponential works in the same way as polar form, multiplication and division are carried out simply by multiplying (for multiplication) or dividing (for division) the coefficients, and then adding (for multiplication) or subtracting (for division) the angle. The complex exponential is important for deriving formulas and is the basis for some of the methods of circuit analysis, but from an algebraic standpoint it behaves in a similar way as polar form. Example: Express a sinusoidal time variation with frequency nu and amplitude A in complex exponential form. What is the angular frequency? Differentiate with respect to time. 5 6 Importance: Fourier analysis and AC Circuits http://www.shef.ac.uk/physics/teaching/phy205/mathematics _for_electromagnetism.htm#solid 2. Partial Differentiation Many physical quantities are a function of more than one variable (e.g. the pressure of a gas depends upon both temperature and volume, a magnetic field may be a function of the three spatial co-ordinates (x,y,z) and time (t)). Hence when differentiating a function there is usually a choice of which variable we differentiate with respect to. For example consider the function f which depends upon the variables x and y (f(x,y)). We can differentiate f with respect to x or y. When we differentiate with respect to a given variable we proceed in the same manner as in basic differentiation for functions which depend upon one variable. However for functions of more than one variable all other variables are treated as if they are constants. The symbols for the differential are modified (‘’ is used instead of ‘d’). The derivatives of f with respect to x or y are written as respectively. Example f(x,y)=3x3+yx2+4xy2+5y3 6 7 We can also take higher order derivatives, e.g. differentiate twice with respect to x or differentiate with respect to one variable (say x ) and then a second (y) Note that for all well behaved functions the order in which we differentiate (e.g. x then y or y then x) is unimportant, i.e. 3. Angles and Solid Angles Angles. We need to consider both normal (onedimensional) and solid (two-dimensional angles) 7 8 In (a) the line of length dl has a component dl cosalong the arc of the circle. The angle is defined as this component divided by the radius of the circle r. Radians. The units are radians. As the circumference of a circle is 2r there are 2r/r=2 radians in a circle. In (b) the area da makes an angle to the surface of the sphere radius r. The projection of da onto the surface of the sphere is hence da cos and the solid angle is defined as this projection divided by the square of the radius The units are steradians. As the area of a sphere is 4r2 there are 4r2/r2=4 steradians in a sphere. 4. Vectors – Vector Analysis Many physical quantities have a direction as well as a magnitude. Examples are force velocity and, in 8 9 electromagnetism, electric and magnetic fields. We describe such quantities using vectors. At each point in space we can imagine an arrow whose length gives the magnitude of the quantity it describes and whose direction corresponds to the direction of the quantity. We want: displacement + angular displacement Components. In dealing with vectors it is often convenient to describe a vector in terms of components. Because space has three-dimensions, three components lying along three orthogonal directions are required to describe any vector. The most common system is the Cartesian one where the three directions are the x, y and z-axes. To define any vector A in the Cartesian system we need the size of the components along the three axes (Ax, Ay and Az) and three unit vectors that are parallel to the three axes (i, j and k parallel to x, y and z respectively). The vector A is given by 9 10 The magnitude of A is given by Non-Cartesian Systems Although the Cartesian system is the most common one and the easiest to visualise and use, There are two other system that are useful when considering problems with cylindrical or spherical symmetry. 10 11 In the cylindrical system the three components of the vector are defined as lying along the radial direction in the x-y plane (r), the angle between the projection onto the x-y plane and the x-axis (f) and the vertical or z component (z). In the spherical system the three components are along the radial direction (r), the angle between the projection onto the x-y plane and the x-axis (f) and the angle between the vector and the z-axis (). Addition of Vectors: associative and commutative Multiplication of Vectors There are two types of multiplication, the dot product, resulting in a scalar and the vector product resulting in a vector. 11 12 The dot product of two vectors (scalar product) If we have two vectors A and B then the dot product of A and B is defined as where A and B are the magnitudes of vectors A and B and is the angle between the two vectors. The dot product is commutative Physically the dot product represents the projection of one of the vectors on to the other times the magnitude of the other. If the two vectors are mutually perpendicular then the dot product is zero (cos 90o=0). In Cartesian co-ordinates, if we have two vectors A and B with components (Ax, Ay, Az) and (Bx, By, Bz) respectively then Physical application of the dot product 12 13 We know from mechanics that if a force F moves through a distance L then the work done is equal to the component of the force along the direction of movement multiplied by L. In the diagram below the component of F along the direction of movement is Fcos. Hence the work done is FLcos . However if we use vectors F and L to describe the force and the movement respectively we have the definition of the dot product. Hence when a force F is moved by a distance L the work done is simply the dot product of the two vectors. This is an application of the dot product which we will use many times in the electromagnetism course. The cross product of two vectors (vector product) If we have two vectors A and B then the cross product of A and B is defined as AB = AB sin n where n is a unit vector normal to the plane containing the two vectors A and B and whose direction is given by 13 14 the right-hand rule. The cross product of two vectors is not commutative as sin(-) = - sin. AB = - BA In Cartesian co-ordinates the cross product of the vectors A and B with components (Ax, Ay, Az) and (Bx, By, Bz) is given by AB = (AyBz-AzBy)i + (AzBx-AxBz)j + (AxBy-AyBx)k The cross product of a vector with itself is zero as the angle between the two vectors is 0 and sin(0)=0. Physical application of the cross product A force F acts at a distance r from a point of rotation. The torque (T) about this point is the distance from where F acts to the point of rotation (r) multiplied by the normal component of F. 14 15 T = rFsin , where is the angle between F and the line drawn through the point of rotation. However if we define torque in terms of a vector whose magnitude gives the size of the torque and whose direction points along the axis of rotation then T= rF where r is the vector from the point of rotation to the point where F acts. The direction of T gives the sense of rotation from the right-hand screw rule. Triple scalar and vector products >>>>>>>>>>>>>>>>> 15 16 5. Calculus of scalars and vectors Much of physics is concerned with how one quantity varies when one or more other quantities change. As in other areas of physics, in electromagnetism we will be concerned with the spatial variation (or derivative) of both scalar and vector quantities. The mathematics can be summarised by the use of a differential operator called 'del' (symbol ) which itself has directional properties. There are three physically meaningful ways in which can be applied to scalars and vectors: it can be applied to a scalar to give a vector (gradient), it can form the dot product with a vector to give a scalar (divergence) and it can form the cross product with a vector to give another vector (curl). Gradient The gradient of a scalar function f is written f or grad f and is given in the Cartesian system by The resultant quantity is a vector. e.g. if f(x,y,z)=2x2+y3+z2xy then 16 17 f= i(4x+z2y) + j(3y2+z2x) + k 2xyz Physical significance of the gradient. At any point the gradient of a function points in the direction corresponding to that for which the function varies most rapidly. The magnitude of the gradient vector gives the size of this maximum variation: the maximum directional derivative. Example: If f(x,y) gives the height (or alternatively the z co-ordinate) of a surface as a function of the x and y coordinates then at any point f will point in the direction of maximum slope of the surface. 17 18 A small ball placed on the surface will tend to roll along the direction opposite to f. The gravitational force acting on the ball is -mgf where m is its mass and g is the acceleration due to gravity. 18 19 The negative sign arises because the force acts in the opposite direction to f. Alternatively if we define U(x,y) as the gravitational potential energy of the ball (U(x,y)=mg f(x,y)) then the gravitational force = -U. This is a general result: Force = -gradient(potential energy). The potential energy may be gravitational, electrical etc. Divergence The divergence of a vector A is written as A or div A and is given by the resultant quantity is a scalar e.g. if A = 3x2yz i + x2z2 j + z2k then ×A=6xyz+2z Sources and sinks. Physical significance. When the divergence of a vector is positive at a given point then there is a source of the vector field at that point. A negative divergence implies a sink for the vector field. We can hence think of the divergence of a vector as telling us how much of the vector field starts (or terminates) at a given point. 19 20 In (a) the vector has a constant magnitude so its divergence is zero. In (b) the x-component increases along the x-direction. This vector hence has a non-zero, positive divergence. Curl The curl of a vector A is written as A or curl A and is given by = The latter is called the determinant. 20 21 The resultant is a vector eg A = yzi-2x2yzj+3x2y2zk A = (6x2yz--2x2z)i+(y-6xy2z)j+(-4xyz-z)k Physical significance. A non-zero curl implies that the corresponding vector field has a sort of rotational property. One way to look for a curl is to imagine that the vector field corresponds to the flow of water. If we place a small paddle wheel in the field then the presence of a non-zero curl suggests that the wheel will rotate. In the above examples for (a) although the field increases along the direction in which it points it produces no rotation of the wheel. However in (b) the field points along x but increases along the y-axis and hence produces a rotation of the wheel. Hence the curl is related to how the field changes as we move across the field. This can also be seen because the expression for curl contains terms Ax/y etc. To some extent curl and div are complementary. The latter requires that the field increases when moving along the field direction, the former that the field increases when moving across the field direction. 21 22 Relationships From the definitions of grad, div and curl the following relationships can be established (f)=0 the curl of a gradient is equal to zero (A)=0 the divergence of a curl is equal to zero The Laplacian: (f)=2f= this is the divergence of a gradient Non-cartesian co-ordinates All of the previous examples are for Cartesian coordinates. For other systems related, but different, expressions exist for grad, div and curl e.g. in cylindrical co-ordinates the gradient is given by For this course you do not need to remember the expressions for non-Cartesian systems but you need to know how to apply them where necessary. 22 23 6. Integration There are two main types of integration for vectors, line and surface Line integrals The line integral of a vector A between the points a and b is given by as we move along a path between the points a and b, at each step we take the component of A which lies along the direction we are moving (given by the vector dl) and multiply it by the distance we move through. The line integral is the sum of all these individual values as we move from a to b. Conservative. In general the path taken between the points a and b must be specified. However for a certain class of vectors the result of the integral is independent of the path taken. Such vectors are said to be conservative. If the line integral is performed around a closed path (initial and final points are the same) then a circle is placed on the integral symbol If the vector A represents a force then the line integral of A between two points gives the work done in moving between these two points. 23 24 Surface integrals The surface integral of the vector A over the surface S is defined as The surface is split into an infinite number of infinitesimally small sections. For each section the product of the area of the section (dS) and the component of A normal to the surface is formed. The integral is the sum of all these products. If the surface is a closed one (no edges) then a circle is placed on the integral sign Relationships between integrals Divergence theorem (Gauss) This states In words 'The surface integral of any vector over a closed surface S is equal to the divergence of that vector integrated over the volume enclosed by S.' Stokes' theorem This states 24 25 In words 'The line integral of any vector around a closed path is equal to the surface integral of the curl of that vector integrated over a surface S which is bounded by the path of the line integral.' THE END 25