Chapter 29 Magnetic Fields
Introduction • Knowledge of Magnetism and application dates back to 13th Century BCE in China. • Magnetic Compass Needle (of Arabic/Indian invention) • Greeks discovered magnetism ~800 BCE. • Magnetite (Fe3O4)attracts iron. • 1269, Pierre de Maricourt discovered that magnets have two poles, N and S, which can attract and repel one another.
Introduction • 1600, William Gilbert suggested the Earth itself is a magnet. • 1750, experimenters collectively show the attractive/repulsive forces follow the inverse square law. • Very similar behavior to electric charges • A single magnetic pole (monopole) has not been isolated.
Introduction • 1819, Hans Christian Oersted, discovered the link between electricity and magnetism. • Current in wire caused deflection of a compass needle. • 1820’s, Michael Faraday and Joseph Henry (independently) showed that a changing magnetic field creates an electric field.
Introduction • 1860’s, James Clerk Maxwell, theoretically shows the reverse. • A changing electric field causes a magnetic field. • This Chapter will focus on the effects of magnetic field on charges and current carrying wires. • We will identify sources of magnetic field in Ch 30.
29.1 Magnetic Fields and Forces • Remember that an electric field surrounds any electric charge. • A magnetic field also surrounds any “moving” charge, and any permanent magnet.
29.1 • Magnetic field is a vector quantity. • Represented by the symbol B • The vector direction of B aligns with the direction that the needle of a compass would point. • Field’s can be represented with magnetic field lines.
29.1 • A typical bar magnetic field lines. • A compass can be used in the presence of a magnet to trace the field lines.
29.1 • Iron filings are also useful (but messy) to identify magnetic field patterns.
29.1 • We can define a magnetic field B, at some point in space, in terms of the magnetic force FB, that the field exerts on a charged particle moving through the space with velocity v. • For now we will assume there are no electric/gravitational fields in the space.
29.1 • Experiments have shown that • The magnitude of FB is proportional the charge q and speed v of the test particle. • The magnitude and direction of FB depend on the velocity and the magnitude and direction of B • When a charged particle moves parallel to the magnetic field vector, the value of FB is zero.
29.1 • Experiments show cont’d: • At any angle θ ≠ 0, FB acts perpendicular to both v and B. • The direction of magnetic force on a positive charge is opposite to the force on a negative charge. • The magnitude of FB is proportional to sinθ
29.1 • We can summarize these observations with the expression
29.1 • Remember the direction of the cross product v x B is determined by the “Right Hand Rule” • Point your fingers in the direction of v. • “Curl” them in the direction of B. • Thumb points in the direction of FB. • Remember the magnitude of a cross product
29.1 • From this we see that FB = 0 when θ= 0, 180o • Electric field Differences • FE acts along field lines, FB perpendicular • FE acts on any particle w/ charge, FB only acts on “moving” charges. • FE does work displacing a charged particle, FB does no work, (perpendicular force)
29.1 • By rearranging the equation we determine units for magnetic field should be • The SI unit for magnetic field is the tesla (T) also given as and
29.1 • Common B Fields
29.1 • Quick Quizzes p. 899 • Example 29.1
29.2 Magnetic Force on a Current Carrying Conductor • If a Force acts on a single charge moving through a B-Field, then it should also act on a wire with current placed in a B-Field. • We can demonstrate this by looking at a wire suspended between the poles of a magnet.
29.2 • When there is no current in the wire, there is no force on the wire.
29.2 • When the current flows upward through the wire, the force causes a deflection to the left.
29.2 • When the current flows down through the wire, there is a deflection to right.
29.2 • To quantify these observations we will look at a wire segment of length L, cross-sectional area A, carrying a current I, in a uniform magnetic field B.
29.2 • We see that the force acting on a single moving charge (traditional current) follows the equation • If we multiple this force by the number of charges moving through the segment given as nAL, we have the force on the whole segment.
29.2 • Now remember that • So we can rewrite the expression as • Where L is a vector that points in the direction of the current and has a magnitude equal to the length of the wire segment.
29.2 • This expression only applies for a straight wire segment passing through a uniform B-Field.
29.2 • Now consider an uniformly shaped, but arbitrarily bent wire, in a B-Field.
29.2 • The force on any small segment will be • We integrate from end points a and b to find the total force on the wire…
29.2 • The quantity represents the vector sum of each small segment, which will equal the vector length from point a to point b, L’
29.2 • So we can conclude that the magnetic force on a curved current carrying wire, in a uniform B-field is equal to that on a straight wire connecting the end points and carrying the same current.
29.2 • We also note that if our conductor is a closed loop, we take the integral over the entire loop • But since the vector sum of a closed loop is zero
29.2 • Quick Quizzes p. 903 • Example 29.2
29.3 Torque on a Current Loop in a Uniform Magnetic Field • Consider the rectangular loop carrying current I in the figure below. • Which sides have the magnetic force acting on them? • Sides 2 and 4 • What directions do the forces act?
29.3 • We see that the magnitude of the forces are equal, • Since the forces act on opposite sides in opposing direction, they have equivalent torque around the central axis.
29.3 • The maximum torque is given as • This can be simplified to
29.3 • If the loop is angled in the B-Field as shown • The lever arm for each torque is • So the overall magnitude of the torque is
29.3 • This can easily be expressed in vector notation as • The direction of vector A points perpendicular to the area of the loop following the RH Rule. • Curl your fingers around the loop in the direction of the current. Thumb Points out A direction.
29.3 • Often the product IA is referred to as the “magnetic dipole moment, μ” or “magnetic moment” of the loop. • The magnetic moment has units of A.m2 and points in the same direction as A.
29.3 • For a coil of wire, with many turns • The potential energy associated with a loop of wire is given as
29.3 • From the expression, Umin = -μB, when μand B point in the same direction and Umax = +μB, when μand B point in opposite directions. • Quick Quizzes p 906 • Example 29.3, 29.4
29.4 The Motion of a Charged Particle in a Uniform Magnetic Field • Remember from 29.1, the magnetic force that acts on a particle is always perpendicular to the velocity (and therefore does no work). • If a positively charged particle moves with velocity v, perpendicular to a magnetic field B, what shape will its path take?
29.4 • Circular Path
29.4 • Using what we know about circular motion, and centripetal acceleration, we can find the radius of the circular path.
29.4 • We can define the angular speed (also called the cyclotron frequency) • The period of the cycle is given as
29.4 • If the velocity is not perpendicular to B, but some arbitrary angle, the perpendicular component causes circular motion, but the parallel component induces no force. • The path is helical.
29.4 • In non-uniform fields the motion is complex. Particles can be come trapped oscillating back and forth.
29.4 • Trapping effect is often referred to as a magnetic bottle. • This effect is shown in the Van Allen radiation belts, surrounding the earth, and is responsible for the Auroras Borealis and Australis .