Magnetic Fields and Forces
Magnetic Fields • Every magnet, regardless of its shape, has two “poles”, called “north” and “south”. • These poles exert forces on each other in a manner analogous to electric charges. • The poles received their names from the behavior of a magnet in the earth’s magnetic field. • Difference: Electric charges can be “isolated” while magnetic poles are always found in pairs.
The Discovery of the Magnetic Field In April of 1820, the Danish physicist Hans Christian Oersted was giving a evening lecture in which he was demonstrating the heating of a wire when an electrical current passed through it. He noticed that a compass that was on the table deflected each time he made the current flow. Until that time, physicists had considered electricity and magnetism to be unrelated phenomena. Oersted discovered that the “missing link” between electricity and magnetism was the electric current.
Hans Christian Oersted (1777- 1851) 3
Serway & Jewett, Principles of Physics, 3rd ed. Figure 22.1
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Current Effect on Compass
Place some compasses around a wire. When no current is flowing in the wire, all compasses point north. When current flows in the wire, the compasses point in a ring around the wire. The Right-Hand Rule: Grip the wire so that the thumb of your right hand points in the direction of the current. Then your fingers will point in the direction that the compasses point. This is the direction of the magnetic field created by the current flow. 6
Vector Conventions
Example: field around a current.
For discussions of magnetism, we will need a three-dimensional perspective, but we will use two-dimensional diagrams when we can. To get the 3rd dimension into a two-dimensional diagram, we will indicate vectors into and out of a diagram by using crosses and dots, respectively. Rule: a dot (•) means you are looking at the point of an arrow coming toward you; a cross (× ×) means you are looking at the tail feathers of an arrow going away from you. 7
The Magnetic Field Definition of the magnetic field: (1) The magnetic field at each point is a vector, with both a magnitude, which we call the magnetic field strength B, and a direction. (2) A magnetic field is created at all points in the space surrounding a current carrying wire. (3) The magnetic field exerts a force on magnetic poles. The force on a north pole is parallel to B, and the force on a south pole is antiparallel to B. 8
Magnetic Field Lines The magnetic field can be graphically represented as magnetic field lines, with the tangent to a given field line at any point indicating the local field direction and the spacing of field lines indicating the local field strength. B-field lines never cross. B-field line spacing indicates field strong strength
weak
B-field lines form closed loops. 9
Two Kinds of Magnetism? We jumped from a discussion of the magnetic effects of permanent bar magnets to the magnetic fields produced by current carrying wires. Are there two distinct kinds of magnetism? No. As we will see in the lectures that follow, the two manifestations of magnetism are actually two aspects of the same fundamental magnetic force.
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Magnetic Fields • Recall: the gravitational field g at some point in space is the gravitational force acting on a “test mass” divided by the test mass, i.e., g = Fg/mo • The electric field E at some point in space is the electric force acting on a “test charge” divided by the test charge, i.e., E = FE/qo
Magnetic Fields • The magnetic field vector B (or “magnetic induction” or “magnetic flux density”) is now defined at some point in space in terms of the magnetic force acting on an appropriate “test object”. • “Test Object” = a charged particle moving with velocity v.
Magnetic Fields q
θ
B v
• The magnitude of the magnetic force, FB , is proportional to the charge q, the speed v = | v | of the particle, and the magnetic field B. • | FB | also depends on θ.
Magnetic Fields • When the charged particle moves parallel to the magnetic field B, then FB = 0. • When the velocity vector v makes an angle θ with the magnetic field B, the magnetic force acts in a direction perpendicular to both v and B. • The magnetic force on a negative charge is in the direction opposite to the force on a positive charge moving in the same direction.
Magnetic Force on a Charge • If the velocity vector makes an angle θ with the magnetic field, the magnitude of the magnetic force is proportional to sin θ. • All of these observations can be summarized by using a special vector notation to write the magnetic force:
FB = q v × B • The product denoted by × is called the cross product.
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Magnetic Force A current consists of moving charges. Ampere’s experiment implies that a magnetic field exerts a force on a moving charge. This is true, although the exact form of the force relation was not discovered until later in the 19th century. The force depends on the relative directions of the magnetic field and the velocity of the moving charge, and is perpendicular to both.
(
F = q v×B
)
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Magnetic Force on Moving Charges Properties of the magnetic force: 1. Only moving charges experience the magnetic force. There is no magnetic force on a charge at rest (v = 0) in a magnetic field. 2. There is no magnetic force on a charge moving parallel (θ θ = 0º) or anti-parallel (θ = 180º) to a magnetic field. 3. When there is a magnetic force, it is perpendicular to both v and B. 4. The force on a negative charge is in the direction opposite to v x B. 5. For a charge moving perpendicular to B (θ θ = 90º) , the magnitude of the force is F=|q|vB.
(
F = q v×B
)
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Magnetic Fields • The direction of the magnetic force FB depends on the sign of the particle’s charge, the direction of its velocity, and on the direction of the magnetic field. • The direction of the magnetic force FB is given by the Right Hand Rule.
Serway & Jewett, Principles of Physics, 3rd ed. Figure 22.4
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Serway & Jewett, Principles of Physics, 3rd ed. Figure 22.3
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Example: Magnetic Force on an Electron A long straight wire carries a 10 A current from left to right. An electron 10 cm above the wire is traveling to the right with a speed of 1x107 m/s. The magnetic field 10 cm away from the wire is 2 x 10-4 T (we will learn to calculate this later) What is the magnitude and direction of the force on the electron.
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Example: Finding Angle Between v and B A proton moving at 4.0 x106 m/s through a magnetic field of 1.7 T experiences a magnetic force of magnitude 8.2x10-13 N. What is the angle between the proton’s velocity and the magnetic field?
Magnetic Fields Differences between electric and magnetic forces on charged particles. • The electric force on a charged particle is independent of the particle’s speed. • The magnetic force only acts on a charged particle when the particle is in motion. • The electric force is always along or opposite to the electric field. • The magnetic force is perpendicular to the magnetic field. (FE = q E vs. FB= q v × B)
Magnetic Fields Differences between electric and magnetic forces on charged particles. • The electric force does work in displacing a charged particle. The magnetic force does no work when a charged particle is displaced. A magnetic field can change the direction but not the speed of a moving charged particle.
Magnetic Fields Units of B The SI unit of B is weber/sq. meter = Wb/m2 = tesla = T = N/(C•m/s ) = 104 G (gauss) Earth’s magnetic field ~ 0.5G = 0.5 x 10-4 T
Magnetic Forces on Current-Carrying Wires When a wire carries a current that is parallel or anti-parallel to a magnetic field, there is no force (because the charges move along field lines). When a wire carries a current that is perpendicular to a magnetic field, there is a force on the wire perpendicular to the current and field.
q q qv I= = = ∆t L / v L
F = qvB = ILB
Fwire = ILB 27
Example: Magnetic Levitation A 0.10 T uniform magnetic field is horizontal, parallel to the floor. A 0.5 m long segment of copper wire with a mass of 0.014 kg is also parallel to the floor and perpendicular to the field. What current through the wire in what direction will allow the wire to “float” in the magnetic field?
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Serway & Jewett, Principles of Physics, 3rd ed. Figure 22.15
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The Force between Two Parallel Wires
Parallel wires carrying current in the same direction attract each other. Parallel wires carrying current in opposite directions repel each other.
Bwire
µ 2I = 0 4π d
We will derive this equation later, but for now it is useful to be aware of it. Finding the force between two wires is a common problem to solve. 31
Ampere’s Experiment When Ampere heard of Oersted’s results, he reasoned that if a current produced a magnetic effect, it might respond to a magnetic effect. Therefore, he measured the force between two parallel current-carrying wires. He found that parallel currents create an attraction between the wires, while anti-parallel currents create repulsion.
André Marie Ampère (1775 – 1836)
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Forces on Current Loops Parallel currents in loops attract.
Opposite currents in loops repel.
Magnetic poles attract or repel because the moving charges in one current producing the pole exert an attractive or repulsive magnetic force on the moving charges in the current producing the other pole. 33
Torques on Current Loops Consider the forces on a current loop carrying current I that is a square of length L on a side that is in a uniform magnetic field B. Its area vector makes an angle θ with B.
Ftop = − Fbottom
Ffront = − Fback
∑F = 0
τ = r × F = rF sin θ
τ = ( F )(d ⊥ ) To find the total torque (net torque), find the torque from each segment and add all the torques together
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τ net = τ left segment + τ right segment
τ net
w w = ( IhB) + ( IhB) 2 2 τ net = ( IBhw)
τ net = ( IAB) 38
If it is not in the position where the torque is maximum, the perpendicular distance changes to:
w d ⊥ = sin θ 2 where θ is the angle between the normal to the loop and the magnetic field. so the net torque is given by:
τ net = τ left segment + τ right segment τ net
w w = ( IhB) sin θ + ( IhB) sin θ 2 2
τ net = ( IBhw) sin θ τ net = ( IAB) sin θ
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If there are multiple loops of wire, then each one contributes the same torque to the loop, so you would multiple the torque of a single loop by the number of loops
τ net = N ( IAB) sin θ
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An Electric Motor We can use the torque of a loop in a magnetic field to make an electric motor. The current through the loop passes through a commutator switch, which reverses the current as the loop approaches the equilibrium position.
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Atomic Magnets A plausible explanation for the magnetic properties of materials is the orbital motion of the atomic electrons. The figure shows a classical model of an atom in which a negative electron orbits a positive nucleus. The electron's motion is that of a current loop. Consequently, an orbiting electron acts as a tiny magnetic dipole, with a north pole and a south pole. However, the atoms of most elements contain many electrons. Unlike the solar system, where all of the planets orbit in the same direction, electron orbits are arranged to oppose each other: one electron moves counterclockwise for each electron that moves clockwise. Thus the magnetic moments of individual orbits tend to cancel each other and the net magnetic moment is either zero or very small. 43
