ELECTRODYNAMICS

Electrodynamics: The Study of Electromagnetic Interactions

• Magnetism is caused by charge in motion.– Charges at rest have just an electric field– But, when they move, they generate both an

electric field and a magnetic field– Can look at individual charges or electric

current in a wire

• Direction of current determines direction of the magnetic field.

• Use right hand rules for analysis.

The Current-Carrying Wire

• Hans Oersted (1820): current-carrying wire generates a magnetic field

• Andre Ampere: magnetic force experienced by a compass near the wire acted at right angles to the current along a series of concentric circles. (Right-Hand-Current Rule)

• Jean Biot & Felix Savart: B-field near a long straight wire is directly proportional to the current and inversely proportional to the distance from the wire

Slide 4

Fig 19.15b, p.678

First Right Hand Rule: thumb points in direction of current, fingers curl in direction of magnetic field- note compass readings. Use for current-carrying wire.

Magnetic field of a long straight wire

• B: magnetic field strength (teslas)• I: current (amperes)• r: radius from wire (meters)• μo: permeability constant in a vacuum• μo = 4π x 10-7 T·m/A

• What is the shape of this magnetic field?

r

IB o

2

Current Loops & Coils• Ampere: field inside loop is much stronger. RHR

provides direction of B. Looks like a dipole field. All magnetism is caused by currents.

• Biot & Savart: field at center of current loop is directly proportional to the current and inversely proportional to the radius of the loop. Stacking several loops overlaps fields and increases field strength. Diameter much greater than thickness.

• Ampere (again): wound wire into long helix (solenoid). With a current passing through solenoid, it acted like a bar magnet. Turn density has a large impact on B-field strength.

For loop or coil of wire, can still use 1st RHR, but direction of current constantly changes.

Easier to use 2nd Right Hand Rule. Fingers curl in direction of current, thumb points to direction of magnetic field.

Magnetic field of a loop of wire carrying current

R

IB o

2

Magnetic Field of Multiple Stacked Loops of Current-Carrying Wire

• The strength of the field is greater than in a single loop.

R

INB o

2

N is the NUMBER of loops

Magnetic Field of a Solenoid

nIB centersolenoid 0,

A solenoid is a helix, so it behaves differently than stacked loops. The major factor for the magnetic field produced by a solenoid is the turn spacing.

n is the turn density, the number of turns per unit length (n = N/L)

At the ends of the solenoid, about half the field “leaks out”.

nIB endssolenoid 021

,

Magnetic Force

• If current exert forces on magnets, then magnets ought to exert forces on currents.

• Ampere:– passed current through two parallel wires– one fixed wire and one suspended to swing

freely– free wire swung in response to B-field

produced by current passing through other wire

Force on a Moving Charge• A charge q moving through a magnetic field (B) with

velocity v, experiences a force, FM, proportional to q, v, and B.

• Only relative motion is necessary. Charge can be at rest and the magnetic field can be moving relative to the charge.

• Direction of the magnetic force is perpendicular to the plane determined by the velocity and B-field vectors.

• Magnitude of FM depends on angle between v & B.• When particle moves parallel to the field, the force is zero.

When the particle moves perpendicular to the field, the force is a maximum.

sinqvBFM

Force on a Moving Charge

3rd Right Hand Rule Version A• Gives direction of the magnetic force exerted on a conventional

current (or positive charge) by an external magnetic field• Point fingers of RH in direction of current (or motion of charge)• Curl fingers through smallest angle to direction of magnetic field• Thumb indicates direction of the force.• If charge or current is negative, direction of force is opposite

(or use left hand).

3rd Right Hand Rule Version B

• Point thumb of RH in direction of current (or motion of charge)

• Straight fingers point in direction of magnetic field

• Palm pushing indicates direction of magnetic force

Magnetic Deflecting Force on a Charged Particle

• Because FM is perpendicular to v, it is a deflecting force.

• It changes the direction of v, without changing the magnitude.

• No work is done on a moving charge by a B-field.

• No change in the particle’s energy will occur in the process.

Magnetic Deflecting Force on a Charged Particle

• If the field is large enough, the direction of the force on the particle will continuously change, but will always be perpendicular to the charge’s velocity.

• The particle will be forced to move in a circular arc (or even a complete circle).

Particles in a Magnetic Field: What is the direction of the particle velocity?

Trajectory of a Free Particle

• The particle experiences a centripetal acceleration.

• The magnetic deflecting force is therefore a centripetal force.

• If v is perpendicular to B, the radius of the charged particle’s trajectory can be easily predicted. qB

mvR

R

mvqvB

R

mvFcent

2

2

momentum

Particle Accelerators• In a particle accelerator, the goal is to obtain the largest possible

momentum, mv.

• This is done by imparting energy to the particle, such as by applying an external E-field. This will increase the momentum of the particle.

• Simultaneously increasing the B-field will keep the radius constant.

• In a particle accelerator, the largest possible field and radius are required for a given charge.

• At Fermilab and CERN, the particle accelerators are 6.3 km and 27 km in circumference, respectively.

Television Screens• Consists of cathode ray tube

(CRT) in which electric fields form a beam of electrons.

• Phosphor on the TV screen glows when struck by beam.

• Pair of coils on the tube neck create a set of perpendicular magnetic fields.

• As the electron beam passes through each set of coils, it is deflected either horizontally or vertically to different regions of the screen.

• The current through the coils can be varied, thereby varying the magnetic field and the degree of deflection.

Click toview animation.

Particles in Magnetic Fields

• All freely accelerating charges radiate electromagnetic energy (we will discuss this in depth later in the year).

• Therefore, a charged particle moving through a magnetic field will lose energy as it experiences a centripetal acceleration.

• If it loses kinetic energy, the radius of its trajectory will decrease and it will spiral inward.

Click toview animation.

Forces on Wires

• Consider a quantity of charge q passing through a wire in a B-field, such that in time t, the charges travels a length of wire l.

• Direction of the force is the same as the direction of the force on the individual positive charge carriers. Use RHR.

sin

sin

sin

sin

IlBF

BtvtqFt

tqvBF

qvBF

tvl

tqI

M

M

M

M

Magnetic Force on Current-Carrying Wire

Force on a Current Loop• Imagine a lightweight current-carrying rectangular coil

placed in uniform B-field.• Direction and magnitude of the force on each segment

of wire depends on the wire’s orientation in the B-field.• Forces may or may not cause the loop to rotate.

No rotation when loop is perpendicular to B-field.

Torque on a Current Loop

• Suspend the loop so it can rotate freely about a vertical axis and place it in a uniform horizontal B-field.

• If the loop is not perpendicular to the field, the forces on the vertical segments of wire produce a torque that rotates the coil through the field.

• The torque on the current loop is given by

where is the angle between the magnetic dipole moment and B.

sinNIAB

The DC motor is simple, yet very important application.

Magnetic Dipole Moment• Tendency of a magnet to

align with external B-field.

• For a planar loop, direction is given by Right-Hand-Current Rule (2nd RHR)

• Magnitude is product of current in the loop and area of the loop.

• Analyzing the torque on the current loop, it is clear why the magnetic moment tends to align with the external B-field.

IAl

Two Parallel Wires

• Two long parallel wires suspended next to each other will either attract or repel depending on the direction of the current in each wire.

• B-field produced by each wire interacts with current in the other wire

• Produces magnetic deflecting force on other wire.

• Wires exert equal and opposite forces on each other.

Two Parallel WiresCurrents in Same Direction…Wires Attract

Currents in Opposite Directions…Wires Repel

Two Parallel Wires

• B-field produced by wire 2 at the position of wire 1 is given by

where d is the distance between the wires.d

IB

2

202

Two Parallel Wires

• Since the wires are parallel, the force exerted by wire 2 on wire 1 is given by

• However, it is often more appropriate to determine the force per unit length on the wire.

d

IIlF

d

IlIF

lBIF

M

M

M

2

2

210

201

21

d

IIBI

l

FM

2

21021