Chapter 29 Magnetic Fields
Magnets • In each magnet there are two poles present (the ends where objects are most strongly attracted): north and south • Like (unlike) poles repel (attract) each other (similar to electric charges), and the force between two poles varies as the inverse square of the distance between them • Magnetic poles cannot be isolated – if a permanent magnetic is cut in half, you will still have a north and a south pole (unlike electric charges) • There is some theoretical basis for monopoles, but none have been detected
Magnets • The poles received their names due to the way a magnet behaves in the Earth’s magnetic field • If a bar magnet is suspended so that it can move freely, it will rotate • The magnetic north pole points toward the Earth’s north geographic pole • This means the Earth’s north geographic pole is a magnetic south pole • Similarly, the Earth’s south geographic pole is a magnetic north pole
Magnets • An unmagnetized piece of iron can be magnetized by stroking it with a magnet (like stroking an object to charge an object) • Magnetism can be induced – if a piece of iron, for example, is placed near a strong permanent magnet, it will become magnetized • Soft magneticmaterials (such as iron) are easily magnetized and also tend to lose their magnetism easily • Hard magnetic materials (such as cobalt and nickel) are difficult to magnetize and they tend to retain their magnetism
Magnetic Fields • The region of space surrounding a moving charge includes a magnetic field (the charge will also be surrounded by an electric field) • A magnetic field surrounds a properly magnetized magnetic material • A magnetic field is a vector quantity symbolized by B • Its direction is given by the direction a north pole of a compass needle pointing in that location • Magnetic field lines can be used to show how the field lines, as traced out by a compass, would look
Magnetic Field Lines • A compass can be used to show the direction of the magnetic field lines
Magnetic Field Lines • Iron filings can also be used to show the pattern of the magnetic field lines • The direction of the field is the direction a north pole would point • Unlike poles (compare to the electric field produced by an electric dipole)
Magnetic Field Lines • Iron filings can also be used to show the pattern of the magnetic field lines • The direction of the field is the direction a north pole would point • Unlike poles (compare to the electric field produced by an electric dipole) • Like poles (compare to the electric field produced by like charges)
Nikola Tesla 1856 – 1943 Magnetic Fields • When moving through a magnetic field, a charged particle experiences a magnetic force • This force has a maximum (zero) value when the charge moves perpendicularly to (along) the magnetic field lines • Magnetic field is defined in terms of the magnetic force exerted on a test charge moving in the field with velocity v • The SI unit: Tesla (T)
Magnetic Fields • Conventional laboratory magnets: ~ 2.5 T • Superconducting magnets ~ 30 T • Earth’s magnetic field ~ 5 x 10-5 T
Direction of Magnetic Force • Experiments show that the direction of the magnetic force is always perpendicular to both v and B • Fmax occurs when v is perpendicular to B and F = 0 when v is parallel to B • Right Hand Rule #1 (for a + charge): Place your fingers in the direction of v and curl the fingers in the direction of B – your thumb points in the direction of F • If the charge is negative, the force points in the opposite direction
Direction of Magnetic Force • The x’s indicate the magnetic field when it is directed into the page (the x represents the tail of the arrow) • The dots would be used to represent the field directed out of the page (the • represents the head of the arrow)
Differences Between Electric and Magnetic Fields • The electric force acts along the direction of the electric field, whereas the magnetic force acts perpendicular to the magnetic field • The electric force acts on a charged particle regardless of whether the particle is moving, while the magnetic force acts on a charged particle only when the particle is in motion • The electric force does work in displacing a charged particle, whereas the magnetic force associated with a steady magnetic field does no work when a particle is displaced (because the force is perpendicular to the displacement)
Force on a Charged Particle in a Magnetic Field • Consider a particle moving in an external magnetic field so that its velocity is perpendicular to the field • The force is always directed toward the center of the circular path • The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle
Force on a Charged Particle in a Magnetic Field • This expression is known as the cyclotron equation • r is proportional to the momentum of the particle and inversely proportional to the magnetic field • If the particle’s velocity is not perpendicular to the field, the path followed by the particle is a spiral (helix)
Chapter 29Problem 3 A proton moves perpendicularly to a uniform magnetic field at 1.0 × 107 m/s and exhibits an acceleration of 2.0 × 1013 m/s2 in the +x-direction when its velocity is in the +z-direction. Determine the magnitude and direction of the field.
Chapter 29Problem 15 A cosmic-ray proton in interstellar space has an energy of 10.0 MeV and executes a circular orbit having a radius equal to that of Mercury’s orbit around the Sun (5.80 × 1010 m). What is the magnetic field in that region of space?
Particle in a Nonuniform Magnetic Field • The motion is complex
Charged Particles Moving in Electric and Magnetic Fields • In many applications, charged particles move in the presence of both magnetic and electric fields • In that case, the total force is the sum of the forces due to the individual fields:
Magnetic Force on a Current Carrying Wire • The current is a collection of many charged particles in motion • The magnetic force is exerted on each moving charge in the wire • The total force is the sum of all the magnetic forces on all the individual charges producing the current • Therefore a force is exerted on a current-carrying wire placed in a magnetic field:
Magnetic Force on a Current Carrying Wire • The direction of the force is given by right hand rule #1, placing your fingers in the direction of I instead of v
Chapter 29Problem 25 A wire having a mass per unit length of 0.500 g/cm carries a 2.00-A current horizontally to the south. What are the direction and magnitude of the minimum magnetic field needed to lift this wire vertically upward?
Magnetic Force on a Current CarryingWire of an Arbitrary Shape • For a small segment of the wire, the force exerted on this segment is • The total force is
Torque on a Current Loop • Applies to any shape loop • Torque has a maximum value when q = 90° • Torque is zero when the field is perpendicular to the plane of the loop
Magnetic Moment • The vector is called the magnetic dipole moment of the coil • Its magnitude is given by μ = IAN • The vector always points perpendicular to the plane of the loop(s) • The equation for the magnetic torque can be written as τ = BIAN sinθ = μB sinθ • The angle is between the moment and the field
Potential Energy • The potential energy of the system of a magnetic dipole in a magnetic field depends on the orientation of the dipole in the magnetic field • Umin = – μB and occurs when the dipole moment is in the same direction as the field • Umax = + μB and occurs when the dipole moment is in the direction opposite the field
Chapter 29Problem 38 A wire is formed into a circle having a diameter of 10.0 cm and placed in a uniform magnetic field of 3.00 mT. The wire carries a current of 5.00 A. Find (a) the maximum torque on the wire and (b) the range of potential energies of the wire-field system for different orientations of the circle.
Answers to Even Numbered Problems • Chapter 29: • Problem 4 • 86.7 fN • 51.9 Tm/s2
Answers to Even Numbered Problems Chapter 29: Problem 46 39.2 mT