THE MAGNETIC FIELD. 9. Magnetism 1

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1 THE MAGNETIC FIELD Magnets always have two poles: north and south Opposite poles attract, like poles repel If a bar magnet is suspended from a string so that it is free to rotate in the horizontal plane, then the end of the magnet that points towards the north geographic pole of the Earth is the north seeking pole Thus the opposite end if the south seeking pole A magnet creates a magnetic field which represents the effect of a magnet on its surroundings, and is represented by the symbol B r The field can be visualised by using small iron filings sprinkled onto a smooth surface placed on top of a bar magnet The iron filings align with the magnetic field in their vicinity 9. Magnetism 1

2 MAGNETIC FIELD LINES Notice that the filings are bunched together near the poles of the magnets This is where the magnetic field is most intense This is demonstrated by drawing field lines that are densely packed near the poles The field weakens further away from the magnet, and the field lines are wider apart The direction of the magnetic field, B r, at a given location is the direction in which the north pole of a compass points when placed at that location Placing a compass near a bar magnet s south pole will cause the north pole of the needle to point toward the south pole of the magnet (opposites attract) Thus the direction of the magnetic field at that location is towards the magnet s south pole As a consequence the magnetic field must point away from the magnet s north pole Magnetic field lines exit from the north pole of a magnet and enter and the south pole 9. Magnetism 2

3 GEOMAGNETISM The Earth produces its own magnetic field, similar to that of a bar magnet, with a pole near each geographic pole The magnetic poles are at an incline to the rotational axis that varies slowly with time by about 11.5 The north pole of a compass needle points toward the north magnetic pole of the Earth Since opposite poles attract it follows that the north geographic pole of the Earth is actually near the south pole of the Earth s magnetic field The Earth s magnetic field lines are horizontal (parallel) near the equator, but enter or leave the Earth vertically near the poles The field is caused by the flowing currents of molten material in the core The field also reverses direction, the last of which was 780,000 years ago 9. Magnetism 3

4 MAGNITUDE OF THE MAGNETIC FORCE ON MOVING CHARGES Here we will consider the force a magnetic field exerts on a moving electric charge Consider a magnetic field B r that points from left to right in the plane of this page A particle of charge q moves through this region with a velocity v r, and the angle between v r and B r is θ Thus the magnitude of the force F r experienced by this particle is given by F = q vbsinθ (Newtons) The magnetic force depends on the charge of the particle q and the magnetic field B It also depends on the speed of the particle v and the magnitude of the angle between the velocity vector and the magnetic field vector, θ A particle must have a charge and must be moving if the magnetic field is to exert a force on it When θ = 0 or 180 the particle moves in the direction of the field, or opposite to it, and the force vanishes The maximum force is experienced when the particle moves at right angles to the field, when θ = Magnetism 4

5 DEFINITION OF THE MAGNITUDE OF THE MAGNETIC FIELD The magnitude, or strength, of a magnetic field is defined by rearranging the previous equation for F B = F/( q vsinθ units 1 N/(A.m) = 1 Tesla (T) 1 gauss = 1G = 10-4 T Example: Particle 1, with a charge q 1 = 3.6µC and a speed v 1 = 862m/s, travels at right angles to a uniform magnetic field. The magnetic force it experiences is N. Particle 2, with a charge q 2 = 53.0µC and a speed v 2 = m/s, moves at an angle 55.0 relative to the same magnetic field. Find the strength of the magnetic field and the magnitude of the magnetic force exerted on particle Magnetism 5

6 DIRECTION OF THE MAGNETIC FORCE Unexpectedly, the magnetic force F r points in a direction that is perpendicular to both B r and v r Figure (a) shows B r and v r lying in the indicated plane The force on a positive charge, F r, is perpendicular to this plane and hence B r and v r The direction of F r can be determined using the right hand rule To find the direction of the magnetic force on a +ve charge, start by pointing the fingers of your right hand in the direction of the velocity v r. Now curl your fingers toward the direction of B r (b). Your thumb points in the direction of F r. If the charge is negative, the force points opposite to the direction to your thumb 9. Magnetism 6

7 MAGNETIC FORCE FOR POSITIVE AND NEGATIVE CHARGES The direction of the magnetic force depends on the sign of the charge The force exerted on a negatively charged particle is opposite in direction to the force exerted on a positively charged particle The magnetic field B r points into the page, and a particle with a positive charge q moves to the right Extend your fingers to the right and curl them into the page and thus from the RHR the force exerted is upward If the charge is negative, the direction of F r is reversed 9. Magnetism 7

8 ELECTRIC VERSUS MAGNETIC FORCES A positively charged particle moving horizontally into a uniform electric field pointing downward experiences a constant downward force (similar to gravity) As a result the particle accelerates downward and follows a parabolic path However were a positive particle were to move in a magnetic field, the magnetic force points into the page As a result, the particle follows a circular path Because the magnetic force is at right angles to the direction of motion, no work is done on it, hence the speed of the particle in a magnetic field is constant Recall that the magnetic force is zero if a particle s velocity is parallel (or antiparallel) to the magnetic field In this case, the particle s acceleration is zero, and thus its velocity remains constant, and will move in a straight line drift along the magnetic field lines 9. Magnetism 8

9 CIRCULAR MOTION (1) Consider a particle of mass m, charge +q, speed v, moves in a region with a constant magnetic field B r pointing out of the page and B r is perpendicular to Thus F = q vbsin90 = q vb At point 1, the particle moves to the right, hence the magnetic force is downward At point 2, it is moving downward, and the magnetic force is to the left At point 3, the force is upwards and so on At each point on the particle s path, the magnetic force is at right angles to the velocity, pointing toward a common centre the condition required for circular motion v r 9. Magnetism 9

10 CIRCULAR MOTION (2) In circular motion, the acceleration is toward the centre of the circle Thus a centripetal force is required to cause the motion In this case, the centripetal force is supplied by the magnetic force Recall that a cp = v 2 /r If ma cp is equal to the magnitude of the magnetic force, then m(v 2 /r) = q vb Thus the radius of circular orbit is r = mv/ q B A mass spectrometer is used to separate isotopes (atoms of the same element with different masses) and to measure atomic masses Inside such a device a beam of ions of mass m and charge +q enter a region of constant magnetic field with a speed v The field causes the ions to move along a circular path, with a radius that depends on the mass and charge of the ion Different isotopes follow different paths and hence can be separated and identified 9. Magnetism 10

11 MASS SPECTROMETER: EXAMPLE Two isotopes of uranium, 235 U (m = kg) and 238 U (m = kg), are sent into a mass spectrometer with a speed of m/s. Given that each isotope is singly ionised and that the strength of the magnetic field is 0.75T, what is the distance d between the two isotopes after they complete half a circular orbit? 9. Magnetism 11

12 HELICAL MOTION Here we consider what happens when a particle has an initial velocity at an angle to the magnetic field In such a case, there is a component of velocity parallel to B r and perpendicular to B r The parallel component of the velocity remains constant with time (zero force in this direction) The perpendicular component results in a circular motion A combination of these motions causes the particle to move in a helical path If the magnetic field is curved (bar magnet, Earth), the helical motion of charged particles will be curved as well 9. Magnetism 12

13 THE MAGNETIC FORCE ON A CURRENT-CARRYING WIRE Whether it travels in a vacuum or a current carrying wire, a charged particle will experience a force when it moves across magnetic field lines A wire with a current will experience a force that is the resultant of all the forces experienced by the individual moving charges Consider a straight wire segment, length L, with a current I flowing from left to right It is in the region of a magnetic field B r at an angle θ to the length of the wire Charges move through the wire at an average speed v The time taken for them to travel L is t = L/v Thus q = I t = IL/v So F = qvb sinθ = (IL/v)vB sinθ = ILB sinθ (Newtons) 9. Magnetism 13

14 THE MAGNETIC FORCE ON A CURRENT-CARRYING WIRE: EXAMPLE A copper rod 0.15m long and with a mass of 0.05kg is suspended from two thin, flexible wires. At right angles to the rod is a uniform magnetic field of 0.55T pointing into the page. Find the direction and magnitude of the electric current needed to levitate the copper rod. 9. Magnetism 14

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