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Chapter 21. Magnetic Forces and Fields. Magnetism Magnetic Force and Field Trajectory of a Charge in a Magnetic Field Magnetic Forces and Torques on Electric Currents Magnetic Fields Produced by Currents Ampere’s Law Magnetic Materials. Chapter 21: Magnetic Forces and Fields. Magnetism.

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### Chapter 21

Magnetic Forces and Fields

Magnetic Force and Field

Trajectory of a Charge in a Magnetic Field

Magnetic Forces and Torques on Electric Currents

Magnetic Fields Produced by Currents

Ampere’s Law

Magnetic Materials

Chapter 21: Magnetic Forces and FieldsMagnetism

The unity of physics:

- heat and thermodynamics kinetic energy, momentum, conservation of energy
- electricity force and energy
- magnetism:
- magnetic fields exert forces on moving charges
- moving charges produce magnetic fields

Magnetism

Magnesia: now called “Manisa,” in western Turkey

Here, around 500 BC, people first noticed that a particular sort of iron ore, suspended from a thread, sought a constant orientation relative to the earth.

Magnetism

No isolated poles (instead, opposed pairs: dipoles)

N & S (“north” and “south”)

Unlike poles attract each other

Like poles repel each other

Pole names result from tendency toward geographic alignment

Magnetism

The “north” pole is the end of a magnetic dipole that seeks the geographic north direction on Earth.

Magnetic Fields -- Qualitative

The magnetic field is a vector that points from a magnetic north pole, toward a magnetic south pole.

The field points in the same direction as the north pole of a compass needle.

Magnetic Fields -- Qualitative

Magnetic field lines point toward a south magnetic pole.

The magnetic field vector is tangent to the field line at each point.

Magnetic Fields -- Quantitative

The magnetic field vector is mathematically defined from its property of exerting a force on a moving charge.

Magnetic Fields -- Quantitative

The magnetic force on a moving charge:

Magnitude:

Direction: given by right-hand rule #1

magnetic field

magnetic force

velocity

charge

Magnetic Fields -- Quantitative

Right-hand rule #1, two ways:

Magnetic Fields -- Quantitative

The magnetic force equation serves as a defining equation for the magnetic field, B:

SI units of magnetic field, B: (tesla)

Other common unit of B: the gauss ( = )

Magnetic Fields and Forces

Properties of the magnetic force on moving charges

- The magnetic force acts only on a moving charge. No force on a stationary charge.
- The magnetic force acts only on a charge whose velocity has a nonzero component perpendicular to B. No magnetic force is exerted if the motion of the charge is parallel to B.
- The magnetic force is perpendicular to the magnetic field.
- The magnetic force does no work on the moving charge.

Magnetic Fields and Forces

The magnetic force does no work on the moving charge?

Trajectory of a Charge in a Magnetic Field

Consider a charged object moving at a velocity v in a plane perpendicular to a uniform magnetic field B:

Trajectory of a Charge in a Magnetic Field

The magnetic force on the charge is always perpendicular to its velocity.

Trajectory of a Charge in a Magnetic Field

At a later time, the force is still perpendicular to the velocity.

Trajectory of a Charge in a Magnetic Field

A force that always acts on an object perpendicular to its velocity is a centripetal force.

This force produces circular motion. (Recall Chapter 5.)

The force, velocity, and path radius are related by:

Trajectory of a Charge in a Magnetic Field

But FC is also the magnetic force:

Mass Spectrometry

If ions have a mass m and a charge q:

Magnetic Force on a Current-Carrying Wire

Recall the magnetic force on a charge moving perpendicular to a magnetic field:

Now, if a current I flows through a straight wire whose length is L:

Magnetic Force on a Current-Carrying Wire

q is now the angle between the field vector B and the wire

The direction of F is still given by right-hand rule #1

(the thumb is now the conventional current)

Torque on a Current-Carrying Loop

A rectangular loop makes an angle f with a magnetic field:

Torque on a Current-Carrying Loop

A rectangular loop makes an angle f with a magnetic field:

Torque on a Current-Carrying Coil

Maximum torque occurs when f = 90° (plane of coil is parallel to B):

Minimum torque occurs when f = 0° (plane of coil is perpendicular to B):

Magnetic Field Produced by a Current

Experimental observation (due to Oersted): a current-carrying wire produces a magnetic field, directly proportional to the magnitude of the current, and inversely proportional to the distance from the wire:

Convert to an equation, using a constant of proportionality:

m0 is called the permeability of free space:

Hans Christian Oersted: 1777 - 1851

Danish physicist; discovered the field due to a current- carrying wire by accident, one evening in April 1820, while preparing for a lecture at the University of Copenhagen

Magnetic Forces Between Two Current-Carrying Wires

Each wire produces a magnetic field at the location of the other wire

Each wire experiences a magnetic force due to the field produced by the other

Magnetic Field Produced by a Current Loop

RHR #2 shows that the magnetic field points out of the plane of the loop on the inside of the loop, and into the plane on the outside of the loop.

The field magnitude is not uniform.

Magnetic Field Produced by a Current Loop

The field magnitude at the center is given by:

If the loop becomes a flat coil, with N turns:

Magnetic Field Produced by a Solenoid

Solenoid: a coil whose length is large compared to its radius.

Field magnitude inside the solenoid:

# of turns per unit length

Ampere’s Law

For an arbitrary closed path about an arbitrary current distribution I:

We can also write this as a sum of “sufficiently” small length elements around the closed path:

Ampere’s Law

Ampere’s Law can be used to calculate the magnetic fields produced by simple current geometries. Example: the long straight wire:

Right-hand rule #2 says that B

is everywhere parallel to Dl. So:

Magnetic Materials

Electrons have orbital and spin rotations in atoms.

These circular motions of charge act as current loops, and produce magnetic dipoles on an atomic scale.

These magnetic dipoles mostly cancel each other. The remaining dipoles tend to be oriented randomly, so bulk materials do not show magnetic dipole behavior.

Magnetic Materials

In ferromagnetic materials (iron, cobalt, nickel, and some alloys), there is a spin coupling of some electrons over small portions of the material called domains.

The domains do behave as magnetic dipoles.

In an “ordinary” object made of a ferromagnetic material, the domains are themselves randomly oriented, and the object is not a magnet.

Magnetic Materials

The ferromagnetic domains may be brought into a common alignment by an external magnetic field (induced magnetism).

In some materials, the common domain alignment is persistent, even when the external field is removed. The object is then a permanent magnet.

The ferromagnetic field can be several orders of magnitude higher than the external field that causes the domain alignment.

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