Concept Summary

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Topic | Discussion | Learning Aids
North and south poles | 21.1 MAGNETIC FIELDS A magnet has a north pole and a south pole. The north pole is the end that points toward the north magnetic pole of the earth when the magnet is freely suspended. Like magnetic poles repel each other, and unlike poles attract each other. |
Direction of the magnetic field | A magnetic field exists in the space around a magnet. The magnetic field is a vector whose direction at any point is the direction indicated by the north pole of a small compass needle placed at that point. |
Magnetic field lines | As an aid in visualizing the magnetic field, magnetic field lines are drawn in the vicinity of a magnet. The lines appear to originate from the north pole and end on the south pole. The magnetic field at any point in space is tangent to the magnetic field line at the point. Furthermore, the strength of the magnetic field is proportional to the number of lines per unit area that passes through a surface oriented perpendicular to the lines. |
Magnetic force (direction) | 21.2 THE FORCE THAT A MAGNETIC FIELD EXERTS ON A MOVING CHARGE The direction of the magnetic force acting on a charge moving with a velocity in a magnetic field is perpendicular to both and . For a positive charge the direction can be determined with the aid of Right-Hand Rule No. 1 (see below). The magnetic force on a moving negative charge is opposite to the force on a moving positive charge. |
Right-Hand Rule No.1 | Extend the right hand so the fingers point along the direction of the magnetic field and the thumb points along the velocity of the charge. The palm of the hand then faces in the direction of the magnetic force that acts on a positive charge. |
| The magnitude B of the magnetic field at any point in space is defined as |
Magnetic field (magnitude) |

| (21.1)

The tesla and the gauss | where F is the magnitude of the magnetic force on a test charge, is the magnitude of the test charge, and v is the magnitude of the charge’s velocity, which makes an angle with the direction of the magnetic field. The SI unit for the magnetic field is the tesla (T). Another, smaller unit for the magnetic field is the gauss; 1 gauss = 10−4 tesla. The gauss is not an SI unit. |
| 21.3 THE MOTION OF A CHARGED PARTICLE IN A MAGNETIC FIELD When a charged particle moves in a region that contains both magnetic and electric fields, the net force on the particle is the vector sum of the magnetic and electric forces. |
A constant magnetic force does no work | A magnetic force does no work on a particle, because the direction of the force is always perpendicular to the motion of the particle. Being unable to do work, the magnetic force cannot change the kinetic energy, and hence the speed, of the particle; however, the magnetic force does change the direction in which the particle moves. |
| When a particle of charge and mass m moves with speed v perpendicular to a uniform magnetic field of magnitude B, the magnetic force causes the particle to move on a circular path of radius |
Radius of circular path |

| (21.2)

| 21.4 THE MASS SPECTROMETER The mass spectrometer is an instrument for measuring the abundance of ionized atoms or molecules that have different masses. The atoms or molecules are ionized (+e), accelerated to a speed v by a potential difference V, and sent into a uniform magnetic field of magnitude B. The magnetic field causes the particles (each with a mass m) to move on a circular path of radius r. The relation between m and B is |
Relation between mass and magnetic field |

Use Self-Assessment Test 21.1 to evaluate your understanding of Sections 21.1, 21.2, 21.3 and 21.4 .

| 21.5 THE FORCE ON A CURRENT IN A MAGNETIC FIELD An electric current, being composed of moving charges, can experience a magnetic force when placed in a magnetic field of magnitude B. For a straight wire that has a length L and carries a current I, the magnetic force has a magnitude of |
Force on a current (magnitude) |

| (21.3)

Force on a current (direction) | where is the angle between the directions of the current and the magnetic field. The direction of the force is perpendicular to both the current and the magnetic field and is given by Right-Hand Rule No. 1. |
| 21.6 THE TORQUE ON A CURRENT-CARRYING COIL Magnetic forces can exert a torque on a current-carrying loop of wire and thus cause the loop to rotate. When a current I exists in a coil of wire with N turns, each of area A, in the presence of a magnetic field of magnitude B, the coil experiences a net torque of magnitude |
Torque on a current-carrying coil |

| (21.4)

| where is the angle between the direction of the magnetic field and the normal to the plane of the coil. |
Magnetic moment | The quantity NIA is known as the magnetic moment of the coil. |
| 21.7 MAGNETIC FIELDS PRODUCED BY CURRENTS An electric current produces a magnetic field, with different current geometries giving rise to different field patterns. For an infinitely long, straight wire, the magnetic field lines are circles centered on the wire, and their direction is given by Right-Hand Rule No. 2 (see below). The magnitude of the magnetic field at a radial distance r from the wire is |
Long, straight wire |

| (21.5)

Permeability of free space | where I is the current in the wire and is a constant known as the permeability of free space . |
Right-Hand Rule No. 2 | Curl the fingers of the right hand into the shape of a half-circle. Point the thumb in the direction of the conventional current I, and the tips of the fingers will point in the direction of the magnetic field . |
| The magnitude of the magnetic field at the center of a flat circular loop consisting of N turns, each of radius R, is |
Center of a circular loop |

| (21.6)

| The loop has associated with it a north pole on one side and a south pole on the other side. The side of the loop that behaves like a north pole can be predicted by using Right-Hand Rule No. 2. |
| A solenoid is a coil of wire wound in the shape of a helix. Inside a long solenoid the magnetic field is nearly constant and has a magnitude of |
Interior of a solenoid |

| (21.7)

| where n is the number of turns per unit length of the solenoid. One end of the solenoid behaves like a north pole, and the other end like a south pole. The end that is the north pole can be predicted by using Right-Hand Rule No. 2. |
| 21.8 AMPÈRE’S LAW Ampère’s law specifies the relationship between a current and its associated magnetic field. For any current geometry that produces a magnetic field that does not change in time, Ampère’s law states that |
Ampère’s law |

| (21.8)

| where is a small segment of length along a closed path of arbitrary shape around the current, is the component of the magnetic field parallel to , I is the net current passing through the surface bounded by the path, and μ0 is the permeability of free space. The symbol indicates that the sum of all terms must be taken around the closed path. |

Use Self-Assessment Test 21.2 to evaluate your understanding of Sections 21.5, 21.6, 21.7 and 21.8 .

Ferromagnetic materials | 21.9 MAGNETIC MATERIALS Ferromagnetic materials, such as iron, are made up of tiny regions called magnetic domains, each of which behaves as a small magnet. In an unmagnetized ferromagnetic material, the domains are randomly aligned. In a permanent magnet, many of the domains are aligned, and a high degree of magnetism results. An unmagnetized ferromagnetic material can be induced into becoming magnetized by placing it in an external magnetic field. |

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