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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

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electrodynamics the study of electromagnetic interactions
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
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
slide4

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.

Fig 19.15b, p.678

Slide 4

magnetic field of a long straight 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?
current loops coils
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.
slide7

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 multiple stacked loops of current carrying wire
Magnetic Field of Multiple Stacked Loops of Current-Carrying Wire
  • The strength of the field is greater than in a single loop.

N is the NUMBER of loops

magnetic field of a solenoid
Magnetic Field of a Solenoid

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”.

magnetic force
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
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 FMdepends 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.
3 rd right hand rule version a
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).
3 rd right hand rule version b
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
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 particle17
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).
trajectory of a free particle
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.

momentum

particle accelerators
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
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.
slide22

Click to

view animation.

particles in magnetic fields
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.
slide24

Click to

view animation.

forces on wires
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.
force on a current loop
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
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 bywhere  is the angle between the magnetic dipole moment and B.

The DC motor is simple, yet very important application.

magnetic dipole moment
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.
two parallel wires
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 wires31
Two Parallel Wires

Currents in Same Direction…

Wires Attract

Currents in Opposite Directions…

Wires Repel

two parallel wires32
Two Parallel Wires
  • B-field produced by wire 2 at the position of wire 1 is given bywhere d is the distance between the wires.
two parallel wires33
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.