Chapter 27, 28 & 29: Magnetism & Electromagnetic Induction

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1 Chapter 27, 28 & 29: Magnetism & Electromagnetic Induction The Magnetic Field The Magnetic Force on Moving Charges The Motion of Charged Particles in a Magnetic Field The Magnetic Force Exerted on a Current-Carrying Wire Loops of Current and Magnetic Torque Magnetic Fields of current carrying wires 1

2 The Magnetic Field The concept of a Field is a convenient way of explaining: Attraction (Force) at a distance. Masses interact via a Field Stationary (static) charges interact via an Field interact via a Field List some of your experiences with magnetic fields: Compass needle aligns with Earth s Magnetic field Attraction and repulsion of Bar magnets Generation of electricity in turbines 2

3 The Magnetic Field Permanent bar magnets have opposite poles on each end, called north and south. Like poles repel; opposites attract. If a magnet is broken in half, each half has two poles: (Can not isolate N & S poles) 3

4 The Magnetic Field The magnetic field can be visualized using magnetic field lines, similar to the electric field. If iron filings are allowed to orient themselves around a magnet, they follow the direction of the field lines. By definition, magnetic field lines exit from the north pole of a magnet and enter at the south pole. Magnetic field lines cannot cross, just as electric field lines cannot. 4

5 The Magnetic Field The Earth s magnetic field resembles that of a bar magnet. However, since the north poles of compass needles point towards the north magnetic pole of the Earth, there is actually a south pole. i.e. the North geographic pole of the Earth is actually near the South pole of the Earth s magnetic field. 5

6 The Magnetic Force on Moving Charges Consider a charge, q, moving with velocity, v, in a magnetic field, B Since, B, v and F all are vector quantities, so a vector cross-product must be introduced. F qv B qvbsin Where is the angle between v and B This is an experimental result we observe it to be true. It is not a consequence of anything we ve learned so far. 6

7 The Magnetic Force on Moving Charges The magnetic force on a moving charge is actually used to define the magnetic field (Magnetic Induction or Magnetic field density): This is similar to Electric field Strength: 7

8 The Magnetic Force on Moving Charges In order to figure out which direction the force is on a moving charge, we use the right-hand rule (RHR). This gives the direction of the force on a positive charge; the force on a negative charge would be in the opposite direction. Point your fingers in the direction of v. Curl your fingers towards B. Then thumb indicates the direction of F. This relationship between the three vectors magnetic field, velocity, and force can also be written as a vector cross product: 8

9 The Magnetic Force on Moving Charges Note that often it is necessary to show a field or a current out of or into the page. The following notation is used: OUT OF PAGE INTO PAGE (i) Examples: Use the RHR to determine the direction of the force acting on the moving charges or current elements shown in the following sketches. (The three axis shown are at right-angles, the apparently angled axis coming out of the page on the left and going in at the right) v v B B B v 9

10 The Magnetic Force on Moving Charges (ii) What is the force if v and B are Parallel (iii) What is the force if q is negative Use of unit vectors i, j and k in 3 Directions j k j -k i i 10

11 Motion of Charged Particles in a Uniform Magnetic Field A positively charged particle in an electric field experiences a force in the direction of the field; In a magnetic field the force is perpendicular to the field. This leads to very different motions: 11

12 Motion of Charged Particles in a Uniform Magnetic Field Question: (a) The sketch shows a particle of mass, m, and charge, q, projected with a velocity, v, on a plane at right-angles to a uniform magnetic field, B. We assume that the speed of the charge is low, therefore we do not need to use relativistic expressions and q B KE Direction of the force: F qv B i k Path: It will travel in a circle of radius r. (b) If a particle with same mass but negative charge and with nine times the energy is injected parallel to the first charge, how will its path differ? It will have a force acting along the j direction and hence travels in the opposite direction with a speed of 3v in a circle of radius 3r. j mv 2

13 The Motion of Charged Particles in a Magnetic Field Because the magnetic force is always perpendicular to the direction of motion, v and field B, the path of a particle is circular. Also, while an electric field can do work on a particle, a magnetic field cannot the particle s speed remains constant. Consider a particle of mass m and charge q, moving at a speed v in a magnetic field B. The particle moves in a circular path of radius, r. F centripetal mv r r 2 mv qb F magnetic qvb 13

14 The Motion of Charged Particles in a Magnetic Field Question: A proton ( m = kg, q = C) moves with a speed of m/s at an angle of 50 to a uniform magnetic field of 38 mt. (I) What is the magnitude of magnetic force on the proton? (ii) What is the acceleration of the proton due to the magnetic field? The operating principle of a mass spectrometer In a mass spectrometer, a beam of charged particles enters a region with a magnetic field perpendicular to the velocity. The particles then follow a circular orbit of radius r = mv/ q B. Particles of different mass will follow different paths. Thus in a mass spectrometer, ions of different mass and charge move in circles of different radii, allowing separation of different isotopes of the same element. 14

15 The Motion of Charged Particles in a Magnetic Field If a particle s velocity makes an angle with the magnetic field, the component of the velocity along the magnetic field will not change A particle with an initial velocity at an angle to the field will move in a helical path. 15

16 The Motion of Charged Particles in a Magnetic Field Exercise: A 0.32 C particle moves with a speed of 16 m/s through a region where the magnetic field has a strength of 0.95 T. At what angle to the field is the particle moving if the force exerted on it is (a) 4.8 x 10-6 N (b) 3.0 x 10-6 N or (c) 1.0 x 10-7 N Exercise: When a charged particle enters a region of uniform magnetic field, it follows a circular path, as indicated in the figure. (i) Is this particle positively or negatively charged? Explain. (ii) Suppose that the magnetic field has a magnitude of T, the particle s speed is 6.0 x 10 6 m/s and the radius of its path is 52.0 cm, Find the mass of the particle, given that its charge has a magnitude of 1.60 x C Give your result in atomic mass units, u, where I u = 1.67 x kg (iii) How long will it take the particle to complete one orbit?

17 The Magnetic Force Exerted on a Current-Carrying Wire The force on a segment of a current-carrying wire in a magnetic field is given by: Where is the angle between and B Since, B, L and F all are vector quantities, so a vector cross-product must be introduced. F I B IBsin

18 Loops of Current and Magnetic Torque In the current loop shown, the vertical sides experience forces that are equal in magnitude and opposite in direction. They do not operate at the same point, so they create a torque around the vertical axis of the loop which causes rotation of the loop. This is the principle used in construction of electric motors. The total torque is the sum of the torques from each force: Or, since A = hw,

19 Loops of Current and Magnetic Torque If the plane of the loop is at an angle to the magnetic field, To increase the torque, a long wire may be wrapped in a loop many times, or turns. If the number of turns is N, we have

20 Loops of Current and Magnetic Torque The torque on a current loop is proportional to the current in it, which forms the basis of a variety of useful electrical instruments. Here is a galvanometer:

21 The Magnetic Force Exerted on a Current-Carrying Wire Exercise: A wire with a current of 2.8 A is at an angle of 36.0 relative to a magnetic field of 0.88 T. Find the force exerted on a 2.25-m length of the wire. Exercise: The magnetic force exerted on a 1.2-m segment of straight wire is 1.6 N. The wire carries a current of 3.0 A in a region with a constant magnetic field of 0.50 T. What is the angle between the wire and the magnetic field? 21

22 Magnetic Fields of current carrying wires One example of the magnetic field is a compass needle placed near a current carrying wire. The needles line up with the direction of magnetic field Experimental observation: Electric currents (moving charges) flowing in wires can create magnetic fields around itself. This is the principle behind the design of electromagnets. These fields form circles around the current carrying wires.

23 Magnetic Fields of current carrying wires To find the direction of the magnetic field due to a currentcarrying wire, point the thumb of your right hand along the wire in the direction of the current I. Your fingers are now curling around the wire in the direction of the magnetic field. The magnetic field is directly proportional to the current in the wire and inversely proportional to the distance from the wire:

24 Magnetic Fields of current carrying wires The magnetic field around a long, straight current carrying wire: T m A 0 = Permeability of free space A general relation between a current-carrying wire of any shape and the magnetic field around itself can be derived using Ampere s Law. (Refer to separate notes on Applications of Ampere s Law)

25 Magnetic Fields of current carrying wires Force between two-parallel wires Since a current-carrying wire experiences a force in a magnetic field, and a magnetic field is created by a current-carrying wire, there is a force between two current-carrying wires:

26 Magnetic Fields of current carrying wires Exercise: Find the magnetic field 6.25 cm from a long, straight wire that carries a current of 5.81 A. Exercise: A long, straight wire carries a current of 7.2 A. How far from this wire is the magnetic field it produces equal to the Earth s magnetic field, which is approximately 5.0 x 10-5 T 26

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