# AP Physics Electromagnetic Wrap Up

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1 AP Physics Electromagnetic Wrap Up Here are the glorious equations for this wonderful section. This is the equation for the magnetic force acting on a moving charged particle in a magnetic field. The angle θ is the angle between the field B and the velocity v, q represents the charge of the particle. If the velocity is perpendicular to the field, then you have a maximum force and it is simply equal to. This is the equation for the magnitude of the force exerted on a straight length of wire by a magnetic field. Where F B is the magnetic force, I is the current, l is the length of the wire, and θ is the angle between the wire and the magnetic field. This is the magnetic field strength around a straight section of a current carrying conductor. B is the magnetic field strength, is the permeability of free space, I is the current, and r is the distance to the center of the conductor. The value for the permeability of free space is: This is the equation for magnetic flux. is the magnetic flux, B is the magnetic field strength, A is the area of the loop, and θ is the angle between B and a normal to the plane of the loop. This is the equation for induced emf. is the induced emf, is the change in magnetic flux, and Δt is the change in time. This is the equation for the induced emf in a loop of width l that is moving into or out of a magnetic field B at a velocity v. 430

2 Here s what you gots to be able to do: Magnetostatics 1. Forces on Moving Charges in Magnetic Field a. You should understand the force experienced by a charged particle in a magnetic field so you can: (1) Calculate the magnitude and direction of the force in terms of q, v, and B, and explain why the magnetic force can perform no work. Okay, this is simply using the old equation. The magnetic force can perform no work because the direction of the magnetic force is always perpendicular to the motion of the particle, so the work is always zero. (2) Deduce the direction of a magnetic field from information about the forces experienced by charged particles moving through that field. This is simply the old right hand rule. (3) State and apply the formula for the radius of the circular path of a charge that moves perpendicular to a uniform magnetic field, and derive this formula from Newton s Second Law and the magnetic force law. To do this, you simply derive the formula for the centripetal force and set it equal to the magnetic force. Solve for the radius. The handout is quite happy to show you how to do this. (4) Describe the most general path possible for a charged particle moving in a uniform magnetic field, and describe the motion of a particle that enters a uniform magnetic field moving with specified initial velocity. The most general path is a helix spiral kind of deal. If the particle s velocity is perpendicular to the field, then the particle will follow a circular path. (5) Describe quantitatively under what conditions particles will move with constant velocity through crossed electric and magnetic fields. The deal here is that the two forces, the electric and magnetic forces, must somehow add up to zero if the particle is to have a constant velocity. So the forces have to be the same magnitude and be in opposite directions. Since the electric force is in the direction of the electric field and the magnetic force is perpendicular to the magnetic field, then the two fields have to be perpendicular to each other. 2. Forces on current-carrying Wires in Magnetic Fields a. You should understand the force experienced by a current in a magnetic field so you can: 431

3 (1) Calculate the magnitude and direction of the force on a straight segment of currentcarrying wire in a uniformed magnetic field Just use the equation; (2) Indicate the direction of magnetic forces on a current-carrying loop of wire in a magnetic field, and determine how the loop will tend to rotate as a consequence of these forces. Use the good old right hand rule. 3. Fields of Long Current-carrying Wires a. You should understand the magnitude field produced by a long straight current-carrying wire so you can: (1) Calculate the magnitude and direction of the field at a point in the vicinity of such a wire. Use the equation. (2) Use superposition to determine the magnetic field produced by two long wires. Superposition means vector addition. Each of the wires produces its own field. You just add them up as vectors. (3) Calculate the force of attraction or repulsion between two long current carrying wires. Use the equation and the equation for the force exerted by a wire,. Electromagnetism 1. Electromagnetic Induction a. You should understand the concept of magnetic flux so you can: (1) Calculate the flux of a uniform magnetic field through a loop of arbitrary orientation. Simply use the equation for magnetic flux. b. You should understand Faraday s Law and Lenz s Law so you can: 432

4 (1) Recognize situations in which changing flux through a loop will cause an induced emf or current in the loop. Faraday s law is simply that a current is induced when the magnetic flux through the conductor is changing. Have to be alert for the idea that it is only when the flux is changing that an emf will be induced. Typical situations where this happens would be where a loop travels into or out of a magnetic field. Please note that once it is in a field, even though it is moving, no emf is induced because the flux is not changing. The other situation is a loop that is rotating within a magnetic field. (2) Calculate the magnitude and direction of the induced emf and current in: (a) A square loop of wire pulled at a constant velocity into or out of a uniform magnetic field. Use the equation rule.. To figure out the current direction, use the right hand (b) A loop of wire placed in a spatially uniform magnetic field whose magnitude is changing at a constant rate. Use the equation. (c) A loop of wire that rotates at a constant rate about an axis perpendicular to a uniform magnetic field. This is a bit more complicated. There are some examples of how to do this in the last electromagnetic handout. Check it out. (d) A conducting bar moving perpendicular to a uniform magnetic field. Just use the good old equation. From 2000: 7. A particle with unknown mass and charge moves with constant speed V = 1.9 x 10 6 m/s as it passes undeflected through a pair of parallel plates, as shown above. The plates are separated by a distance d = 6.0 x 10-3 m, and a constant potential difference V is maintained between them. A uniform magnetic field of magnitude B = 0.20 T directed into the page exists both between the plates and in a region to the right of them as shown. After the particle passes into the region to the right of the plates where only the magnetic field exists, its trajectory is circular with radius r = 0.10 m. 433

5 (a) What is the sign of the charge of the particle? Check the appropriate space below. Positive _xx_ Negative Neutral It cannot be determined from this information. If the particle was positive, the magnetic force would be up (right hand rule) and the particle would curve above the plates, it s motion is the opposite, so the particle must have a negative charge. Between the plates, the negative particle is deflected downward. Therefore the electric field must force the negative particle up, but direction of the field is the direction a positive test charge would go so the field must be down. (b) On the diagram above, clearly indicate the direction of the electric field between the plates. (c) Determine the magnitude of the potential difference V between the plates. The magnetic field strength is given by: Magnetic force: Electric Force: Set the electric force and the magnetic force equal to one another: (d) Determine the ratio of the charge to the mass ( q/m) of the particle. This sounds bad. Okay, the particle follows a curved path in the magnetic field. So we can set the centripetal force = magnetic force. Solving for q/m 434

6 From 1999: A rectangular conducting loop of width w, height h, and resistance R is mounted vertically on a nonconducting cart as shown above. The cart is placed on the inclined portion of a track and released from rest at position P 1 at a height y o above the horizontal portion of the track. It rolls with negligible friction down the incline and through a uniform magnetic field B in the region above the horizontal portion of the track. The conducting loop is in the plane of the page, and the magnetic field is directed into the page. The loop passes completely through the field with a negligible change in speed. Express your answers in terms of the given quantities and fundamental constants. (a) Determine the speed of the cart when it reaches the horizontal portion of the track. (b) Determine the following for the time at which the cart is at position P 2 with one-third of the loop in the magnetic field. i. The magnitude of the emf induced in the conducting loop ii. The magnitude of the current induced in the conducting loop 435

7 (c) On the following diagram of the conducting loop, indicate the direction of the current when it is at position P 2. (d) i. Using the axes below, sketch a graph of the magnitude of the magnetic flux through the loop as a function of the horizontal distance x traveled by the cart, letting x = 0 be the position at which the front edge of the loop just enters the field. Label appropriate values on the vertical axis. ii. Using the axes below, sketch a graph of the current induced in the loop as a function of the horizontal distance x traveled by the cart, letting x = 0 be the position at which the front edge of the loop just enters the field. Let counterclockwise current be positive and label appropriate values on the vertical axis. 436

8 From 1996: A rigid rod of mass m and length is suspended from two identical springs of negligible mass as shown in the diagram below. The upper ends of the springs are fixed in place and the springs stretch a distance d under the weight of the suspended rod. (a) Determine the spring constant k of each spring in terms of the other given quantities and fundamental constants. But each spring holds ½ the total mass As shown below, the upper end of the springs are connected by a circuit branch containing a battery of emf ε and a switch S, so that a complete circuit is formed with the metal rod and springs. The circuit has a total resistance R, represented by the resistor in the diagram below. The rod is in a uniform magnetic field, directed perpendicular to the page. The upper ends of the springs remain fixed in place and the switch S is closed. When the system comes to equilibrium, the rod is lowered an additional distance Δd. (b) What is the direction of the magnetic field relative to the coordinate axes shown on the right in the previous diagram? Using the right hand rule, the field is out of the page, along the positive z axis. The current is to the right thumb points that way and the bar is pushed down, so the palm points down. The fingers end up pointing out of the page. (c) Determine the magnitude of the magnetic field in terms of m,, d, Δd, ε, R, and fundamental constants. Gravity initially displaced the bar a distance d. Then the magnetic field displaced the bar an extra distance, Δd. Set the spring force equal to the magnetic force: But, there are two springs 437

9 x is Δd, so plug that in: Solve for B: Now we need to get B in terms of the items listed in (c): If the internal resistance is not mentioned, we can assume that it is zero. So Plug this into the equation we developed for I: Also, we already found a value for Plug this into the equation for k: (d) When the switch is suddenly opened, the rod oscillates. For these oscillations, determine the following quantities in terms of d, Δd, and fundamental constants: i. The period. But there are two springs. Plug in the value for k. ii. The maximum speed of the rod. 438

10 Use conservation of energy: The 2 in front of U S because there are two springs and From 1995: In a linear accelerator, protons are accelerated from rest through a potential difference to a speed of approximately 3.1 x 10 6 meters per second. The resulting proton beam produces a current of 2.0 x 10-6 ampere. a. Determine the potential difference through which the protons were accelerated. The electric potential energy is equal to the work done on the particle. This must equal its kinetic energy. Note, we can look up the mass of a proton (it is 1.67 x kg). b. If the beam is stopped in a target, determine the amount of thermal energy that is produced in the target in one minute. 439

11 Because energy is conserved, the thermal energy produced in the target is simply equal to the power of the beam of particles. The proton beam enters a region of uniform magnetic field B, as shown above, that causes the beam to follow a semicircular path. c. Determine the magnitude of the field that is required to cause an arc of radius 0.10 meter. d. What is the direction of the magnetic field relative to the axes shown above on the right? Positive Z -- out of the paper.. Use the right hand rule. Thumb points in direction of velocity, palm points in direction of magnetic force, and the fingers point in the direction of the field. At the dot on the curved path of the proton, the palm points to the left towards the center of the circle. The thumb points down (the velocity at that point is a tangent to the circle and is down). Your fingers end up pointing out of the page. 440

12 From 1994: A force F is applied to a conducting rod so that the rod slides with constant speed v over a frictionless pair of parallel conducting rails that are separated by a distance. The rod and rails have negligible resistance, but the rails are connected by a resistance R, as shown below. There is a uniform magnetic field B perpendicular to and directed out of the plane of the paper. a. On the following diagram, indicate the direction of the induced current in the resistor. Using the right hand rule, the palm must point to the left (the direction of the force which opposes the motion of the bar), the fingers point in the direction of the field out of the page. This thumb then points in the direction of the current through the bar, which is down. Therefore the current is clockwise and the current through the resistor is up. Determine expressions for the following in terms of v, B,, and R. b. The induced emf in the rod. c. The electric field in the rod. d. The magnitude of the induced current in the resistor R. 441

13 We know that the emf induced is: We also know that, if the resistance in the rod and rails is negligible, then the emf is equal to the voltage. So: Now we can apply Ohm s law. e. The power dissipated in the resistor as the rod moves in the magnetic field. f. The magnitude of the external force F applied to the rod to keep it moving with constant speed v. At constant speed the net force must equal zero. From 1998: The long, straight wire shown in Figure 1 below is in the plane of the page and carries a current I. Point P is also in the plane of the page and is a perpendicular distance d from the wire. Gravitational effects are negligible. a. With reference to the coordinate system in Figure 1, what is the direction of the magnetic field at point P due to the current in the wire? 442

14 Use the right hand rule. Into the page, in the minus z direction. A particle of mass m and positive charge q is initially moving parallel to the wire with a speed v o when it is at point P, as shown in Figure 2 below. b. With reference to the coordinate system in Figure 2, what is the direction of the magnetic force acting on the particle at point P? Right hand rule. Force is down, in the minus y direction. Thumb points in direction of v 0, fingers point into page (direction of field), so the palm points down, the direction of the force. c. Determine the magnitude of the magnetic force acting on the particle at point P in terms of the given quantities and fundamental constants. d. An electric field is applied that causes the net force on the particle to be zero at point P. i. With reference to the coordinate system in Figure 2, what is the direction of the electric field at point P that could accomplish this? The electric field direction is defined as the direction a positive particle moves. The positive particle must move up to counter the downward magnetic field. ii. Determine the magnitude of the electric field in terms of the given quantities and fundamental constants. 443

15 From 1992: The figure shows a cross section of a cathode ray tube. An electron in the tube initially moves horizontally in the plane of the cross section at a speed of 2.0 x 10 7 meters per second. The electron is deflected upward by a magnetic field that has a field strength of 6.0 x 10-4 Tesla. a. What is the direction of the magnetic field? Out of the page. b. Determine the magnitude of the magnetic force acting on the electron. c. Determine the radius of curvature of the path followed by the electron while it is in the magnetic field. An electric field is later established in the same region as the magnetic field such that the electron now passes through the magnetic and electric fields without deflection. d. Determine the magnitude of the electric field. 444

16 e. What is the direction of the electric field? The force needs to be downward on the electron to cancel the upward magnetic field force. Unlike the magnetic field which is perpendicular, the electric field is parallel to its force. The electric field is the direction a positive charge moves. Electrons move opposite the field. So if the electron needs to move down the electric field is up. From 1993: A particle of mass m and charge q is accelerated from rest in the plane of the page through a potential difference V between two parallel plates as shown. The particle is injected through a hole in the right-hand plate into a region of space containing a uniform magnetic field of magnitude B oriented perpendicular to the plane of the page. The particle curves in a semicircular path and strikes a detector. Neglect relativistic effects throughout this problem. a. Determine the following i. State whether the sign of the charge on the particle is positive or negative. Positive. The right plate is negative and the left is positive. The particle is attracted to the negative plate and repelled by the positive plate so it has to be positive. ii. State whether the direction of the magnetic field is into the page or out of the page. Out of the page. Look at the particle at the mid point of the curve (where the little arrowhead is located). Its velocity is down, so point your thumb down. It is being forced towards the center by the magnetic force, so point your palm to the left. This leaves your fingers pointing out of the page, which is the direction of the magnetic field. b. Determine each of the following in terms of m, q, V, and B. i. The speed of the charged particle as it enters the region of the magnetic field B. ii. The force exerted on the charged particle by the magnetic field B. 445

17 iii. The distance from the point of injection to the detector. iv. The work done by the magnetic field on the charged particle during the semicircular trip. No work is done since the magnetic field is perpendicular to the displacement. 446

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