Physics 142 Mathematical Notes Page 1. Mathematical Notes

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1 Physics 142 Mathematical Notes Page 1 Mathematical Notes This set of notes contains: a review of vector algebra, emphasizing products of two vectors; some material on the mathematics of vector fields; some important mathematical approximations; and a discussion of Euler s formula, which will be used in our discussion of electromagnetic waves. t the end are some worked-out examples.! Vector lgebra In ordinary three dimensional space a vector is defined as a collection of three numbers which depend on the directions of the coordinate axes being used. When those axes are rotated to new directions, the numbers change. The rules for those changes are the same as those that apply to the coordinates of a point in pace. In fact, the prototype vector is one that specifies a particular point in space. If an object is at that point, this vector is called its position vector, usually denoted by r. In printed material one denotes a vector by a bold face! letter, such as. In handwriting various notations are used, such as an arrow over it ( ), or underlining it (). The numbers constituting the vector are usually given in reference to a cartesian coordinate system, so they are denoted by subscripts: x, y, z. These numbers are called the components of the vector. One can specify the vector in terms of its components. The simplest way is just to list them: = ( x, y, z ). For example, the position vector would be written as r = (x, y, z). The magnitude of the vector is a positive number, denoted in print simply as the same letter in italics (). It is defined in terms of the components as follows: = x 2 + y 2 + z 2. One often reads that a vector is a quantity with magnitude and direction. Specifying a direction needs two angles and a coordinate system to define them. These and the magnitude are still three numbers. Vector equations. Two vectors are equal if and only if all three components are correspondingly equal. = B means: x = B x, y = B y, z = B z. Multiplication by a scalar. scalar is a quantity that does not change when the coordinate axes are rotated; it has no direction associated with it. Examples are ordinary numbers, time, mass, and energy. The magnitude of a vector is a scalar. vector can be multiplied by a scalar; one simply multiplies each component by the scalar. The resulting quantity is another vector. If c is a scalar the vector equation B = c means three equations: B = c means: B x = c x, B y = c y, B z = c z. One sees easily that the for the magnitudes we have B = c.

2 Physics 142 Mathematical Notes Page 2 ddition of vectors. To add two vectors one simply adds the components. The result is another vector. If C = + B then we have three equations: C = + B means: C x = x + B x, C y = y + B y, C z = z + B z. (For subtraction, just replace + by in all these equations.) Representation by arrows. convenient graphical way to represent a vector is to draw it as an arrow. The length of the arrow represents the magnitude, and the direction (relative to some coordinate system) is specified by which way the arrow points. This is particularly useful when dealing with situations in two dimensions because all the arrows lie in one plane, so they can be drawn on a page.. Shown is the arrow representing a vector in 2-D. The components shown are easily obtained from the right triangle: x = cos, y = sin. lthough the arrow shown here starts at the coordinate origin, that is necessary only for position vectors. Other arrows can be moved around as needed, as long as the length and direction are maintained. rrows can be used to represent the addition of two vectors. Shown is an example, where C = + B. The two arrows representing the vector being added are placed head-to-tail. The arrow representing the sum starts at the tail of the first and ends at the head of the second. To represent subtraction, use the fact that multiplying a vector by 1 reverses its direction, so the corresponding arrow is reversed. Unit vectors. common way to represent a vector is to use a set of three unit vectors, usually called i, j, k. These have unit magnitude and are directed along the x, y, and z coordinate axes respectively. Then one writes = x i + y j+ z k. This allows the three components to be written on one line, which is useful when writing equations. Sometimes one introduces other unit vectors to represent directions. For example, to specify only the direction of the position vector r one introduces the unit vector ˆr = r/r. The caret over the symbol denotes a unit vector. Multiplication of vectors. If one multiplies each component of = ( x, y, z ) by each component of B = (B x,b y,b z ), one gets a set of 9 numbers. This set of products by itself does not change in simple ways when the coordinate axes are rotated. But combinations of these 9 can be found that have simpler properties. One combination is a scalar therefore unchanged by rotation of the coordinate axes. nother set of three combinations turns out to be a vector. These are useful in the description of nature. Scalar product. The scalar product of the two vectors and B is defined in terms of their components as follows B = x B x + y B y + z B z. y x C y x B

3 Physics 142 Mathematical Notes Page 3 One uses the dot multiplication sign, so this is also called the dot product. From the definition it is clear that B = B, i.e., this multiplication is commutative. In arrow representation the scalar product is conveniently written in terms of the magnitudes and relative direction of the vectors. In the case shown we have a very useful way to write it: B = Bcos. B To prove this, let the vectors lie in the x-y plane with in the +x-direction. Then x =, y = 0 while B x = Bcos, B y = Bsin. So B = x B x = Bcos as claimed. Since the product is a scalar, this is its value regardless of how we choose the coordinate axes. Some properties of the scalar product: The maximum value (when = 0 ) is B. The minimum value (when = π ) is B. If the vectors are perpendicular ( = π /2 ) the product is zero. If B = then we find = 2. This gives another definition of the magnitude of a vector, and shows why it is a scalar. n important use of the scalar product is in the discussion of energy and work, as given in the review notes on energy. Vector product. nother set of combinations of the products of components of two vectors gives a quantity that is another vector. We write this as C = B, which is defined as follows: C x = y B z z B y C = B means: C y = z B x z B z C z = x B y y B x Because the cross multiplication sign is used, this is also called the cross product. One sees that B = B, so it is important to specify which vector in the product is first. Because the result is a vector, to use the arrow representation requires giving rules for both the magnitude and the direction of the result. Take the case shown in the drawing above: we find for the magnitude C = B = Bsin. To prove this, use the same coordinate system as above. We see that C z = x B y = Bsin and all other components of C are zero. The magnitude of C is the same as C z (Because is limited to be between 0 and π, sin is never negative.) So C = Bsin as claimed. In the process we have also shown that the direction of C is perpendicular to that of both and B. The direction of C in the arrow representation is determined by a right hand rule : 1. Place the arrows representing and B tail to tail.

4 Physics 142 Mathematical Notes Page 4 2. Curl the fingers of your right hand from the direction of to the direction of B. 3. The direction of C, perpendicular to both and B is indicated by your thumb. Some properties of the vector product: If the two vectors are along the same line, the product is zero ( sin = 0 ). If the vectors are perpendicular to each other, the magnitude is C = B, which is the largest it can be. For the standard Cartesian unit vectors we have i j = k, j k = i, k i = j. Flux and Circulation of Vector Fields This course centers around the concept of a field, a physical quantity distributed in space which has a specific value at each point. The temperature in a room is a field. Often a field in physics is defined by giving a method of measuring its value at a given point. For temperature one could imagine placing a thermometer at each point and recording its reading. The values of the field can be scalars (like temperature) or vectors (like the velocity of flow at the various points in a fluid). The gravitational field g is a vector field. The field g is defined as the gravitational force per unit mass. If one places a mass m at a point where the field value is g the gravitational force on it will be mg. So the field g is what we often call the gravitational acceleration. There are two integral quantities that are properties of a vector field which we will use in this course. They are called flux and circulation. Flux of a vector field. The prototype vector field is the velocity field in a fluid, and the terminology derives from that case. n important aspect of the flow of a fluid is the amount of material that passes in unit time through unit area, a quantity called the flux. This depends on the velocity of the flow, on the mass density of the flowing material and on the relative orientation of the velocity and the surface through which it flows. Shown is the situation in two views. The vector V represents the field. The vector has magnitude equal to the area of the surface and is directed perpendicular (normal) to the surface. The angle between the directions of these vectors is. The flux Φ is defined as Φ = V. Because of the scalar product, the flux is largest when = 0 (when the surface is normal to the flow). It is zero if V is perpendicular to (when the flow is along the surface). For non-uniform fields and arbitrary surfaces calculation of the flux involves an integral. We define the infinitesimal flux through an infinitesimal bit of surface by dφ = V d and integrate over the surface to find the total flux: V V

5 Physics 142 Mathematical Notes Page 5 Φ = V d. This integral is intrinsically multi-dimensional. We will calculate it only in cases of simple geometry and simple field behavior, but it is important to have a geometrical understanding of it. If the surface is closed (completely surrounds a volume) the integral for the flux is denoted by a special symbol:! V d. Two of the four basic equations of the electromagnetic field involve this kind of flux integral. Circulation of a vector field. The circulation is the line integral of a field along a closed path:! V dr. This line integral is of the type used to calculate the work done by a force. If the circulation of the field is zero, the field is called conservative, by analogy with the definition of a conservative force. The other two basic equations of the electromagnetic field involve the circulation. Mathematical pproximations In this course there will be many situations where mathematical approximations are used to obtain simple formulas for special cases. Especially important are the binomial approximation and the small angle approximations. Binomial approximation. Consider the function f (x) = (1 + x) n, where x is a small number compared to 1, and n is any real number. One can use a Taylor series to expand f (x) as a series in powers of x. Because x is small, the successive term in the series will be smaller and smaller, and the first few terms are often a good approximation. Recall the formula for the Taylor series: f (x) = f (0) + f (0) x + 1 2! f (0) x n! f (n) (0) x n + where the primes denote derivatives. We will use only the first two terms as our approximation, for which we need f (x) = (1 + x) n, so f (0) = 1 ; f (x) = n(1 + x) n 1, so f (0) = n. The resulting approximation is (1 + x) n 1 + nx. This is the binomial approximation, useful if x << 1. It will be used often in this course. If a better approximation is needed, perhaps because terms linear in x cancel, the third term in the Taylor series can be included.

6 Physics 142 Mathematical Notes Page 6 It often happens that the expression at hand is not in the form (1 + x) n, but something like (a + x) n, where x << a. In this case we do a little algebra first and then use the approximation: (a + x) n = a n (1 + x/a) n a n (1 + nx/a). Small angle approximations. For angle measured in radians, there are (Taylor) power series representing the trigonometric functions: sin = 3 3! + 5 5! 7 cos = 1 2 2! + 4 4! 6 7! + 6! + These converge for 0 2π. If is no larger than about 0.1 (about 6 ), a good approximation can be obtained by using only the first terms: sin, cos 1, tan. These small angle approximations will be used in this course, especially in optics. In some cases the approximation for the cosine may require using the first two terms of the series. Exponentials and logarithms. Similar Taylor expansions exist for these functions: e x = 1 + x + x2 x2 +, ln(1 + x) = x 2! 2 + x3 3 + For small values of x keeping only the first two terms gives a good approximation: e x 1 + x, ln(1 + x) x x2 2. Euler s formula. Comparing the series given above, one can derive directly a remarkable result: e i = cos + isin where is an angle (in radians) and i = 1. Changing to, we have a second formula (the complex conjugate of the first): e i = cos isin. These can be combined to give formulas we will use in wave optics. cos = ei + e i, sin = ei e i. 2 2i

7 Physics 142 Mathematical Notes Page 7 Sample problems and solutions. 1.! n airplane flies to its destination in two steps. First it flies due east 100 miles; then it flies at angle 37 north of east for another 100 miles. a.! b.! To fly straight to the destination, at what angle north of east does it go? How far will it travel? a.! Call east the x-direction and north the y-direction. Let the vectors representing the two steps of the trip be r 1 and r 2, and call the vector representing the straight-line trip R. Then R = r 1 + r 2, so the problem is one of adding vectors. The figure shows the situation. dd the components: R x = r 1x + r 2x = cos37 = 100( ) = 180 R y = r 1y + r 2y = sin 37 = = 60 y R r 2 37 r 1 x! We find from the right triangle with legs R x and R y : tan = R y /R x = 60/180 = 1/3, or b.! We also find R = R x 2 + R y 2 = (180) 2 +(60) mi. 2.! Sunlight striking the earth is represented by a vector S. Its magnitude S is the intensity, the power the beam transports across unit area perpendicular to the beam; the direction of S is the direction of energy flow. To represent a bit of the earth s surface we use a vector. Its magnitude is the area of that bit, and its direction is normal (perpendicular) to the surface. The situation is as shown. The total power impinging on the surface is the flux of S through the surface area: P = S.! t the summer solstice the sun is directly overhead the Tropic of Cancer at noon, so the angle is 0. t the equinox that angle is Find the ratio of the power impinging on the surface at equinox to that at solstice. From the figure we see that P = Scos. The other factors do not change, so the ratio is P equinox = cos P solstice cos0 This shows one reason for the seasonal changes in temperature. The other reason is the change in the length of the daylight period. Both of these effects are larger for latitudes farther from the equator.

8 Physics 142 Mathematical Notes Page 8 3.! s will be discussed, the force acting on a particle of electric charge q moving with velocity v in a region where there is a magnetic field B is given by the formula F = qv B. Consider a proton from the sun, with positive charge e, entering the magnetic field of the earth, as shown in the figure. Find the magnitude and direction of the force. The magnitude of the force is e times the magnitude of v B, so it is F = evbsin. The direction is that of v B (because the charge is positive). Use the right hand rule. First put the arrows tail to tail, then curl the fingers of the right hand from v toward B. The thumb points into the page, which is the direction of the force. You may have to do some twisting of your wrist to get the fingers going the right way. It s important to curl them from the first vector in the product toward the second, not the other way around. B v 4.! In the notes is calculated the electric field for a circular flat disk with charge spread uniformly over it. The field along the symmetry axis, at distance x from the disk, is shown to be: E x (x) = 2kQ R 2 1 x. x 2 + R 2! Here k is a universal constant, Q is the total charge on the disk, and R is its radius. Use the binomial approximation to show that far from the disk (for x >> R ) this expression reduces to the formula for the field of a point charge: E x (x) = kq x 2. We will work with the second term in the [ ], writing it as x x 2 + R 2 ( ) 1/2. First factor out the large term: ( x 2 + R 2 ) 1/2 = x 1 ( 1 + R 2 /x 2 ) 1/2. The x 1 cancels the factor of x. The expression is now in proper form to apply the binomial approximation, since R 2 /x 2 is much smaller than 1. We find 1 + R 2 /x 2 ( ) 1/2 1 (1/2) R 2 /x 2. Putting this into the original formula, we see that the 1s cancel, leaving us with the expression we want. This result is an example of a general property. If you are far away from all the charges in a given system, and if the total charge Q of the system is not zero, the electric field at your location is approximately that of a point charge Q. If the total charge is zero, a better approximation is needed.

9 Physics 142 Mathematical Notes Page 9 5.! n object rests on the bottom of a pool of water, of depth d. n observer looks down from above the water and sees the object, but it appears to be below the surface by a smaller distance d. Explain why and determine the ratio d /d. The situation is shown in the drawing. Two light rays emanate from the object, indicated by the arrows. One ray goes straight up and is not refracted (bent) at the surface. The other starts out at angle from the vertical and is refracted at the surface, emerging at angle. The observer s eyes interpret the rays as having originated at a point at distance d below the surface. d d x The distance along the surface between these rays is x. There are two right triangles with opposite side x. From the taller one we find x/d = tan, and from the shorter one x/ d = tan. Therefore we have d /d = tan /tan. Now we introduce some physics, the law of refraction, which relates the two angles. This law (which will be discussed in the course) is usually written as nsin = n sin, where the ns are indices of refraction, measures of the speed of light in the media. The definition is n = c/v, where c is the speed of light in a vacuum and v is its speed in the medium. For the air above the surface we have n 1, so we can write the law as nsin = sin, or sin /sin = 1/n. ir, like most gases, does not appreciably slow down light, so its index is very close to 1. Now we use the fact that if these two rays are both to enter through the pupil of the observer s eye the angle between them must be very small. We can use the small angle approximations for both angles: tan sin (in radians, of course). This gives us the answer to the original question: d /d / 1/n. For water n 4/3, so the object appears to be only about 3/4 as deep as it actually is.