E-Physics Tutorial Book

Note: I have attached below the lesson for the 3 activities. Thank you so much for your time and effort.

Activity 1.

Directions: Use the right-hand rule to determine the direction of the force given the direction of the magnetic field and the velocity of the particle. Draw the answer with proper label. Thank you.

Activity 1. Written Work: A chargedvparticle is placed in a magnetic field. Use the right - hand rule to determine the direction of the force given the direction of the magnetic field and the velocity of the particle. Draw your answer with proper label. Activity 2. Directions: Answer the question below. What are the two conditions that must be satisfied so that a charge will experience a force when it is placed in a magnetic field? Condition 1: Condition 2: Activity 3. Directions: Please indicate what is given, required, and solution to each of the problems below. A proton moves through a uniform magnetic field of magnitude 2.0 plat a speed of 5.00 x '10"3 m/s. What is the magnitude of the maximum force it can experience? Required: A proton moving horizontally at 3.5 sz enters a uniform magnetic field of 0.075l' directed upward at 300 with the horizontal. Find the magnitude of the magnetic force acting on the proton. Required: Magnets and Magnetic Fields 1.1 Magnetism We have observed a magnet attract paper clips, nails and other objects made of iron. Any magnet, whether it is in the shape of a bar or a horseshoe, has two ends or faces, called poles, which is where the magnetic effect is strongest. If a bar magnet is suspended from a fine thread, it is found that one pole of the magnet will always point toward the north. It is not known for sure when this fact was discovered., but it is known that the Chinese were making use of it as an aid to navigation by the eleventh century and perhaps earlier. This is the principle of a compass. A compass needle is simply a bar magnet which is supported at its center of gravity so that it can rotate freely. The pole of a freely suspended magnet that points toward geographic north called the north pole of the magnet. The other pole points toward the south and is called the south pole. Before the relationship of magnetic interactions to moving charges was understood, the interactions of permanent magnets and compass needles were described in terms of magnetic poles. If a bar-shaped permanent magnet, or bar magnet, is free to rotate, one end points north. This end is called a north pole or N pole; the other end is a south pole or S pole. Opposite poles attract each other, and like poles repel each other (Fig.1). An object that contains iron but is not itself magnetized {that is, it shows no tendency to point north or south) is attracted by either pole of a permanent magnet (Fig. 2}. This is the attraction that acts between a magnet and the unmagnetized steel door of a refrigerator. By analog: to electric interactions, we describe the interactions in Figs. 1 and 2 by saying that a bar magnet sets up a magnetic eld in the space around it and a second body responds to that eld. A compass needle tends to align with the magnetic field at the needle's position. (a) Opp-mill; palm. attract. (3) IN , E. (b) IJLC pulxs- lapel. Motion of Charge Particles in Electric and Magnetic Fields We have seen that when a charged particle moves within a magnetic eld, a force acts on the particle perpendicular to the velocity. When a force acts perpendicular to the velocity, that force is not able to do any work. Thus, the force does not increase nor decrease the speed of the particle. Instead, it can deect its motion such that it moves on an are or even a circle. The resulting motion is called uniform circular motion for which an object moves 0 8 0 9 around in a circular path at a constant speed. Let us say that a positively charged particle is moving at a velocity v to the right as shown in Fig. 6. And 0 0 then, the magnetic eld is turned on such that it is perpendicular to the particle's velocity. 0 0 Figure 6. A charged parade moving perpendicular to a magnetic fieid moves around in a circle. Photo credit: Breaking Through (".1'nr-2ru-3rml Physics 2 . . The result is that the path of the particle gets deected into a circle of radius R that is dependent on how fast the particle is moving. Since 8 = 90, the magnitude of the magnetic force is: :33 = M (Eq. 1) This is the force that accelerates the particle. From Newton's second law, the centripetal force Fe is: mvz F= E.2 c R (Q) Since FB = F5, we get: We simplify this and solve for R. U R: 5.3 (pg (9') In this equation, R is the radius of the circular path, m is the mass of the particle, v is the particle's speed, q is the absolute value of the charge of the particle, and B is the magnitude of the magnetic eld. The result of using this equation is the same for both positive or negative charges. The only difference is whether the particle turns in a counterclockwise direction as shown. If the particle is negative, then the force is in the opposite direction; the particle moves clockwise. 3.1 Frequency in Circular Motion Now that the charged particles is going around in circles, it undergoes periodic motion. Its frequency id constant. Recall that frequency is the number of cycles that an object completes in every second. From its definition, frequency f is expressed in hem (Hz), which is the same as cycles per second (cps). It is related to the angular frequency cu (also known as angular velocity} of the object which is defined by the equation: a; = 2m\" (Eq. 4) where the 21: changes the unit to radians per second (rad/ s). the angular velocity is related to the linear frequency by the radius: 1; = rm We will use Eq. 3 again and replace the v above in terms of linear frequency: _ mv _ mrw _ qB _ qB The radius will cancel out, then we can simplify and isolate the angular velocity: q3 = E .5 w m ( q ) From this equation, we can conclude that for a charged particle moving in circles in a magnetic eld, the frequency depends only on the fundamental properties of the particle: its charge and mass. We can change its frequency of rotation simply by changing the magnetic eld. Increasing the magnetic eld makes the particle go around faster. Plugging in Eq. 4 into Eq. 5 will give us: = (Eq. 6) This is called the cyclotron frequency. The cyclotron is a device invented by ED. Lawrence and M.S. Livingston in 1934 to accelerate particles such as protons to high kinetic energies by moving them around and around in circles. Equation 6 shows that the frequency of the particles inside a cyclotron is independent of the radius of the path. Example A proton (m = 1.67x10'2?kg) is moving in circles of radius 15 cm in a transverse magnetic eld of magnitude 2500 G. Find the following: 1. Frequency of the motion 2. Speed of the proton Solution: We are given the mass of the proton. We also know its charge, 1.6x10'196. Other given details are r = 0.15 m and B = 2500 G = 0.25 T. 1. The frequency of the proton is given by Eq. 6. qt? _ (1.6x10'19)(0.25 T) _ 3 31 106B _21rm_ 21r(1.67x10'27kg) _ ' x Z 2. The speed of the proton can be found using Eq. 3. We just isolate v: _ qBR _ (1.6x10'196)(0.25 T)(0.15m) m 1.6'7x10'2l'kgr v = 3.59x106m/s 3.2 The Velocity Selector In Figure 7, a positively charged particle enters a region wherein there is an electric eld and a magnetic eld. The velocity of the particle, which is horizontally to the right, is perpendicular to the two elds. In other words, the particle's velocity, magnetic eld, and electric eld are all mutually perpendicular to each other. As a result, the particle experiences both an electrostatic force and a magnetic force. If the electric eld is directed downward, the force FE on the positive charge is also downward. The magnetic eld is directed into the paper so that the magnetic force ngs directed upward. The two elds can be adjusted independently until the two forces cancel out. The particle is then in equilibrium and it moves in a straight line. The first condition of equilibrium gives us: FB = F E qu = qE The q cancels out and we can get v: E E 7 :1 '2 Figure 7. A positively charged particle moves perpendicular to both the magnetic eld and the electric eld. The resulting forces are shown on the right. Photo credit: Breaking Through General Physics 2 The diagram shown in Fig. 7 is called the velocity selector because the ratio E/ B is related to the velocity of particles. When adjusted correctly, only those particles moving at the corresponding velocity can move through. For the rest of the particles, the magnetic force and electrostatic force will not quite cancel out. Every time there is a resultant force, the path of the particles bends away and gets ltered out. Only those particles that travel in a straight line reach the end of the path. 3.3 Thomson's Discovery Long before the electron was discovered, physicists used a device called a cathode ray tube or CRT to fire a beam called the cathode ray toward the screen labeled S in Fig. 8. J .J . Thomson suspected that particles were being emitted by the CRT. Thomson accelerated the particles across a potential difference V, giving them kinetic energy to cross the length of the CRT, striking the screen on the front of the tube. This potential difference is related to the velocity v of the particles. K = 11 1 2 _ V 21111; 3 Thus, the particles' speed is: _ 28V E 8 v m ( q- ) We combine Eq. 8 with Eq. 7 and we will get: E_ 23V B- m from which we can get the charge-to-mass ratio i: 5'2 23V B2_m e F2 m 2VB2 Figure 8. Schematic of a cathode ray tube, or the picture tube in old TV sets and computer monitors. Photo credit: Breaking Through General Physics 2 Remarkably, Thomson found that no matter what E, V, and B are, the ratio of e/m is constant. We know that this makes sense because both charge and mass are constant. However, remember that this was a time when it was believed that the atom was the smallest unit of matter. Later, Thomson would be honored for discovering the electron, the first elementary particle. Thomson's experiment provided no way to find the charge and the mass independently. It would be 15 years before Robert A. Millikan would determine the charge of the electron precisely using another experiment (the famous "milk drop" experiment). Magnetic Force on a Current-Carrying Wire Lesson So far, we have established that a charge moving freely in a magnetic field experiences a force. If we put a current-carrying wire in a magnetic field, the free charges in the wire have velocity as well. So, if we put the wire in a magnetic field, then the wire as a whole should experience a force too. In Figure 9, we see a wire exposed to a perpendicular magnetic field. In (B), there is a magnetic field but no current. In (C), the current is directed upward. It twitches to the left as if there is a force directed toward the left. In (D), the current direction is changed. Now the force also changes direction. A B DFigure 9. What happens when a wire is piaced in a magnetic eld? Photo credit: Breaking Through General Physics 2 Inside the wire are charge carriers. As usual, let us pretend that the charge carriers are positive. In Figure 10, we take a close look at a straight segment of current-carrying wire placed in a magnetic eld B directed into the paper. The current I is toward the right, due to the charges moving to the right. a e s a L 0 g 93 -- Il-l ' , " 0 0 8 3 0 G . I Figure 10. A straight segment of wire carrying a current white in a magnetic eld B Photo credit: Breaking Through General Physics 2 ' Figure 11. Magnetic force on a conducting wire Photo credit: Breaking Through General Physics 2 To calculate the force exerted on the wire, consider a segment of wire of length 4? and cross-sectional area A, as shown in Figure 1 1. The magnetic eld E" points into the page, and is represented with crosses (X). The charges move at an average drift velocity 1301. Since the total amount of charge in this segment is th = q(nA1','), where n is the number of charges per unit volume, the total magnetic force on the segment is F}, = Qm a, x E = ani'(d x E?) = 192:1?) where I = nqdA, and if? is a length vector with a magnitude i' and directed along the direction of the electric current. In Figure 10, the charged particles have a velocity v to the right. So, we know that there should be a force equal to: FB = qv x B And when wen use the RHR, we know that this force is upward. How many charges are actually moving along the wire? To nd out, we introduce the constant n, the number of charges per unit volume. This is a property of the metal. If we multiply n by the volume Vof the wire, we can get the number of charges. To get V, we take a small piece of wire of length L (Fig. 10). Multiplying this length by the cross-sectional area A of the wire will give us its volume. Therefore, the total force on all the charges in that piece is equal to: FB = (nAL)qv x B (Eq. 1) where the nAL term gives us the number of charges within that small volume. Recall that the charges crossing an area per unit time is equal to the current. The product nAqu actually defines the current in the wire. If we substitute / in Eq. 1, we get: FB = IL x B (Eq. 2) Note that the original equation is a vector product, so Eq. 2 should also be a cross product of two vectors. In this case, the length L of the wire, is a vector, whose direction is the same as that of the current. What we arrived at is an equation for the force that a current-carrying wire experiences when placed in a magnetic field. It is a general equation and can be used no matter the shape of the wire is. And since it is a cross product of the vectors L and B, we can determine the direction of the force using the RHR. In doing so, follow the same rules as for v x B, but point your fingers in the direction of the current. A composite sketch of the magnetic circles is shown in Figure 12, where the field strength is shown to decrease as you get farther from the wire by loops that are farther separated. Magnetic field lines (a) (b) Figure 12. (a) When the wire is in the plane of the paper, the field is perpendicular to the paper. Note the symbols used for the field pointing inward (like the tail of an arrow) and the field pointing outward (like the tip of an arrow). (b) A long and straight wire creates a field with magnetic field lines forming circular loops. Photo credit: Physics Libretexts In case there will be an integration of an angle in Eq. 2, this will be the given as FB = IL x B sin 0 (Eq. 3) Which represents the force F that is proportional to the length L of the wire, current I in the wire, magnetic field B, and the angle 0 between the current and the magnetic field. The maximum value of the sine function is 1. If the current is perpendicular to the magnetic field, 0 = 90 and sin 90 = 1. The force must be a maximum. If the current is parallel to the field, 0 = 0 and sin 0 = 0. The force is also zero. A charge q moving with velocity v in a magnetic field also experiences a force F. Suppose that a charge covers distance L in time t.Recall that I = q/t Replacing L by vt and I by q/t in the Equation 3 (F8 =11. xB sin 9), the force on a moving charge is given by FB = qu sin 9. Example A 100 m long power line stretching in a north-south direction has current I=200m A directed north. If Earth's magnetic eld at that location is 0.4 T directly downward, what is the 3 North magnetic force acting on the . power line? Eta I Figure 12. Power line Photo mdit: Breaking Through General Physics 2 Solution: We are given the length L = 100 m of the wire, I = 2000 A, and its direction to the north. We also know that B = 0.4 T, downward. From Figure 12, we can see that the angle 0 between I and B is 90. The magnitude of the magnetic force is: F}, = [L x B sinB = (2000 A)(100 m)(0.4 T) sin90 = 3x104N Now we apply the RHR. Using your right hand, point your ngers toward the right of the page. Make sure your palm is facing downward so you can curl your ngers toward B. 1Which way is your thumb pointing? It is pointing toward the paper, so the wire experiences a force toward the west. Some applications of electromagnetism One major goal of physics is the study of how an electric eld can produce an electric force on a charged object. A closely related goal is the study of how a magnetic eld can produce a magnetic force on a (moving) charged particle or on a magnetic object such as a magnet. You may already have a hint of what a magnetic eld is if you have ever attached a note to a refrigerator door with a small magnet or accidentally erased a credit card by moving it near a magnet. These magnet acts on the door or credit card via its magnetic field. The applications of magnetic elds and magnetic forces are countless and changing rapidly every year. Here are just a few examples. For decades, the entertainment industry depended on the magnetic recording of music and images on audiotape and videotape. Although digital technologr has largely replaced magnetic recording, the industry still depends on the magnets that control CD and DVD players and computer hard drives; magnets also drive the speaker cones in headphones, TVs, computers, and telephones. A modern car comes equipped with dozens of magnets because they are required in the motors for engine ignition, automatic window control, sunroof control, and windshield wiper control. Most security alarm systems, doorbells, and automatic door latches employ magnets. In short, you are surrounded by magnets. The science of magnetic elds is physics; the application of magnetic elds is engineering. Both the science and the application begin with the question \"What produces a magnetic el A Figure 1. (a) Two bar magnets attract when opposite poles (N and Figure 2. (a) Either pole of a bar magnet S, or S and N) are next to each other. attracts an unmagnetized object that contains b) The bar magnets repel when like iron, such as a nail. (b) A real-life example of poles (N and N, or S and S) are next this effect. to each other. Photo credit: University Physics with Photo credit: University Physics with Modern Modern Physics 13th edition Physics 13th edition Earth's Magnetic Field The earth itself is a magnet. Its north geographic pole is close to a magnetic south pole, which is why the north pole of a compass needle points north. The earth's magnetic axis is not quite parallel to its geographic axis (the axis of rotation), so a compass reading deviates somewhat from geographic north. This deviation, which varies with location, is called magnetic declination or magnetic variation. Also, the magnetic field is not horizontal at most points on the earth's surface; its angle up or down is called magnetic inclination. At the magnetic poles the magnetic field is vertical. Figure 3 is a sketch of the earth's magnetic field. The lines, called magnetic field lines, show the direction that a compass would point at each location. The direction of the field at any point can be defined as the direction of the force that the field would exert on a magnetic north pole. North geographic pole The geomagnetic north pole is actually (earth's rotation axis) a magnetic south (S) pole-it attracts the N pole of a compass. Figure 3. A sketch of the earth's magnetic Compass field. The field, which is -Magnetic field lines show caused by currents in the direction a compass the earth's molten core, would point at a given changes with time; location. geologic evidence The earth's magnetic shows that it reverses field has a shape direction entirely at similar to that pro- irregular intervals of duced by a simple bar magnet (although 104 to 106 years. actually it is caused by electric currents in the core) Photo credit: The earth's magnetic axis is University Physics with offset from its geographic axis. Modern Physics 13th edition The geomagnetic south pole is actually a South geographic pole magnetic north (N) pole. 1.2 Magnetic Field To introduce the concept of magnetic field properly, let's review our formulation of electric interactions in the previous lessons, where we introduced the concept of electric field. We represented electric interactions in two steps: 1. A distribution of electric charge at rest creates an electric field E in the surrounding space. 2 . The electric field exerts a force F = qE on any other charge q that is present in the field.We can describe magnetic interactions in a similar way: 1. A moving charge or a current creates a magnetic field in the surrounding space (in addition to its electric field). 2 . The magnetic field exerts a force F on any other moving charge or current that is present in the field. Like electric field, magnetic field is a vector field-that is, a vector quantity associated with each point in space. We will use the symbol B for magnetic field. At any position the direction of B is defined as the direction in which the north pole of a compass needle tends to point. The arrows in Fig. 3 suggest the direction of the earth's magnetic field; for any magnet, points out of its north pole and into its south pole The Definition of B We determined the electric field E at a point by putting a test particle of charge q at rest at that point and measuring the electric force FF acting on the particle. We then defined E as E FE (Eq. 1) If a magnetic monopole were available, we could define B in a similar way. Because such particles have not been found, we must define B in another way, in terms of the magnetic force FB exerted on a moving electrically charged test particle. Moving Charged Particles. In principle, we do this by firing a charged particle through the point at which B is to be defined, using various directions and speeds for the particle and determining the force FB that acts on the particle at that point. After many such trials we would find that when the particle's velocity v is along a particular axis through the point, force FB is zero. For all other directions of v, the magnitude of FB is always proportional to v sino, where o is the angle between the zero-force axis and the direction of v. Furthermore, the direction of FB is always perpendicular to the direction of v. (These results suggest that a cross product is involved.) The Field. We can then define a magnetic field B to be a vector quantity that is directed along the zero-force axis. We can next measure the magnitude of FB when is directed perpendicular to that axis and then define the magnitude of B in terms of that force magnitude: FB B = lalv' (Eq. 2) where q is the charge of the particle. We can summarize all these results with the following vector equation: FB = qu x B; (Eq. 3)that is, the force FB on the particle is equal to the charge q times the cross product of its velocity v and the field B (all measured in the same reference frame). Using the cross product, we can write the magnitude of FB as FB = IqlvB sin d, (Eq. 4) where o is the angle between the directions of velocity v and magnetic field B. SI Unit for Magnetic Field The SI unit for that follows from Eq. 3 and 4 is the newton per coulomb-meter per second. For convenience, this is called the tesla (T): newton 1 tesla = 17 = 1- meter (coulomb) (second Recalling that a coulomb per second is an ampere, we have newton N 1T =1 (coulomb) (second) A . m An older name for the tesla is the "weber per meter squared" (1 2 = IT) An earlier (non-SI) unit for B, still in common use, is the gauss (G), and 1 tesla = 104 gauss 1G = 10-4T A field given in gauss should always be changed to teslas before using with other SI units. 1.3 Electric Field vs. Magnetic Field Electric Field Magnetic Field O Byjus.com Figure 4. An Electric field and Magnetic field lines Photo credit: Byjus.com An object with a moving charge always has both magnetic and electric fields. They have some similarities and also have two different fields with the same characteristics. Both fields are inter-related called electromagnetic fields, but they are not dependent on each other. The magnetic field is an exerted area around the magnetic force. It is obtained by moving electric charges. The direction of the magnetic field is indicated by lines. While the electric fields are generated around the particles which obtainelectric charge. During this process, positive charges are drawn, while negative charges are repelled. The area around a magnet within which magnetic force is exerted, is called a magnetic field. It is produced by moving electric charges. The presence and strength of a magnetic field is denoted by "magnetic flux lines". The direction of the magnetic field is also indicated by these lines. The closer the lines, the stronger the magnetic field and vice versa. When iron particles are placed over a magnet, the flux lines can be clearly seen. Magnetic fields also generate power in particles which come in contact with it. Electric fields are generated around particles that bear electric charge. Positive charges are drawn towards it, while negative charges are repelled. A moving charge always has both a magnetic and an electric field, and that's precisely the reason why they are associated with each other. They are two different fields with nearly the same characteristics. Therefore, they are inter-related in a field called the electromagnetic field. In this field, the electric field and the magnetic field move at right angles to each other. However, they are not dependent on each other. They may also exist independently. Without the electric field, the magnetic field exists in permanent magnets and electric fields exist in the form of static electricity, in absence of the magnetic field. 1.3a. What are Electric and Magnetic Fields? Magnetic fields are created whenever there is a flow of electric current. This can also be thought of as the flow of water in a garden hose. As the amount of current flowing increases, the level of magnetic field increases. Magnetic fields are measured in milliGauss (mG). An electric field occurs wherever a voltage is present. Electric fields are created around appliances and wires wherever a voltage exists. You can think of electric voltage as the pressure of water in a garden hose - the higher the voltage, the stronger the electric field strength. Electric field strength is measured in volts per meter (V/m). The strength of an electric field decreases rapidly as you move away from the source. Electric fields can also be shielded by many objects, such as trees or the walls of a building. 1.3b. Nature An electric field is essentially a force field that's created around an electrically charged particle. A magnetic field is one that's created around a permanent magnetic substance or a moving electrically charged object. 1.3c. Movement In an electromagnetic field, the directions in which the electric and magnetic field move, are perpendicular to each other 1.3d. Units The units which represent the strengths of the electric and magnetic field are also different. The strength of the magnetic field is represented by either gaussor Tesla. The strength of an electric field is represented by Newton per Coulomb or Volts per Meter. 1.3e. Force The electric field is actually the force per unit charge experienced by a non moving point charge at any given location within the field, whereas the magnetic field is detected by the force it exerts on other magnetic particles and moving electric charges. However, both the concepts are wonderfully correlated and have played important roles in plenty of path breaking innovations. Their relationship can be clearly explained with the help of Maxwell's Equations, a set of partial differential equations which relate the electric and magnetic fields to their sources, current density and charge density. Difference Between Electric Field and Magnetic Field Electric Field Magnetic Field Nature It creates an electric charge Creates an electric charge in surrounding around moving magnets Units Measured as newton per Measured as gauss or tesla coulomb Force Proportional to the electric Proportional to charge and charge speed of electric charge Movement in Perpendicular to the Perpendicular to the electric Electromagnetic field field magnetic field Measuring An electric field is measure The magnetic field is device is measured using an measured using the electrometer magnetometer 1.4 Magnetic Flux Magnetic flux is a measure of the number of magnetic field lines passing through an area (the product of the average magnetic field times the perpendicular area that it penetrates). The symbol we use for flux is the Greek letter capital phi, The equation for magnetic flux is: D = BA cose, where 0 is the angle between the magnetic field B and the area vector A . The area vector has a magnitude equal to the area of a surface, and a direction perpendicular to the plane of the surface. The SI unit for magnetic flux is the weber (Wb). 1 Wb = 1 T m2. Faraday's law states that an induced current is produced whenever the flux changes. The flux depends on the magnetic field B, area A, and the angle 0. A change in any of these three factors constitutes a change in flux.Fig. A. To maximize the magnetic flux through a flat (a) (b) area, orient the area so the plane of the area is perpendicular to the direction of the magnetic field. (a) shows a perspective view, while (b) shows the view looking along the field lines. In this case, the area vector is in the same direction as the field lines. (Full faced area perpendicular to the magnetic field gives a maximum magnetic flux) Fig. B. There is no flux when the plane of the area (a) (b) is parallel to the field. (a) shows a perspective view, while (b) shows the view looking along the field lines. In this case, the area vector is perpendicular to the field lines. (An area must be perpendicular to the magnetic field) Fig. C. Tilting the loop from the orientation in Figure (a) (b) A reduces the flux. (a) shows a perspective view, while (b) shows the view along the field lines. (Lesser exposure of surface area means lesser flux) Example The figure below is a perspective view of a flat surface with area 3.0 cm2 in a uniform magnetic field B. The magnetic flux through this surface is +0.90 mWb. Find the magnitude of the magnetic field and the direction of the area vector A. (a) Perspective view (b) Our sketch of the problem (edge-on view) 120 Figure D. (a) A flat area A in a uniform magnetic field B (b) The area vector A makes a 60 angle with B. (If we had chosen A to point in the opposite direction, 0 would have been 120 and the magnetic flux PB would have been negative.) Solution Identify and Set Up: Our target variables are the field magnitude B and the direction of the area vector. Because Bis uniform, B and 0 are the same at all points on the surface. Hence, we can use PB = BA cose Execute: The area A is 3.0x10-4m; the direction of A is perpendicular to the surface, so 0 could be either 60 or 120. But PB, B, and A are all positive, so cose must also be positive. This rules out 120, so 0 = 60, see figure above. Hence, we find0.90 x10 3Wb B = A cos 0 (3.0x 10-4m2) (cos 60) Evaluate: In many problems we are asked to calculate the flux of a given magnetic field through a given area. This example is somewhat different: It tests your understanding of the definition of magnetic flux. 2 Lorentz Force Lesson Lorentz force is defined as the combination of the magnetic and electric force on a point charge due to electromagnetic fields. It is used in electromagnetism and is also known as the electromagnetic force. The force that a magnetic field exerts affects another material at the atomic level. Recall that an electric field exerts a force of FE = qE (Eq. 1) on a charge. This is the electrostatic force. Magnetic fields also exert forces on a charged particle but only when that charged particle is moving. The magnetic force FB is defined as the cross product: axab = ad (Eq. 2) In this equation, q is the charge, vis the velocity of the charge, and B is the external magnetic field. To evaluate the cross product, we need to know the angle 0 between u and B. Then the magnitude of the magnetic force is: FB = qVB sin 0 (Eq. 3) The quantity B is a measure of the strength of the magnetic field. It is also known as magnetic flux density or magnetic induction. The SI unit of magnetic field is the tesla (T), named after the Serbian inventor, Nikola Tesla. A tesla is the amount of magnetic field that will exert a force of 1 N on a charge of 1 C moving at 1 m/s perpendicular to that field. The equivalent units are: 1T = m Another commonly used but not nonstandard unit of magnetic field is the gauss (G), named after Carl Friedrich Gauss, known for formulating Gauss's law. Remembering to convert any magnetic field given in gauss to tesla when solving problems. The conversion is: 1G = 10-4TThe magnetic eld of refrigerator magnets is in order of about 100 gauss. The magnetic field of Earth in some places is about 0.5 G. It is possible for a charged particle to experience both an electrostatic force and a magnetic force at the same time. The net force is given as: F=qE+qva (Eq.4) This known as the Lorentz force, named after Dutch physicist Hendrik Antoon Lorentz. The Right-Hand Rule Sometimes, when we multiply two vectors, we get a scalar. This is accomplished by taking the dot product of the vectors, and one familiar example of this is work. Work is the dot product or scalar product of force and displacement. At other times, when we multiply two vectors, the product is also a vector. The magnetic force is an example. Once the result of Eq. 3 is calculated, the direction is needed. For this, we will use the right-hand rule (RHR). First, we need a way to draw three-dimentional problems. To draw the problem, we are going to use the notations O and . Think of a vector, such as force, as an arrow (Fig. 5). It is easy enough to draw arrows when they are at on a paper. But when an arrow is going into the paper, imagine that you are looking at the backend of the arrow, and you will see something like this: . For an arrow coming out of the paper is its tip, and then we draw this: 0. After the vectors are drawn, label the angle 9 which is the smaller angle between I) and B. Figure 5. (a) An arrow is used to represent vectors. (b) The righthand rule. Photo credit: Breaking Through General Physics 2 To apply the right-hand rule (RHR), follow these steps: 1. With your right hand, point you ngers in the direction of v (the first vector in the cross-product 1) x3). 2. Curl your hand toward B (the second vector in v x B). Your ngers should sweep the angle 8. 3. Which way is your thumb pointing? That is the direction of the force is if the charge is positive. If the charge is negative, follow the same steps, but then ip the direction 180