Two Chapter 6

Magnetic Forces on Conductors and Moving Charges

• Basic Law of Magnetism - Two magnetic fields will interact to produce forces of attraction (opposite poles) and repulsion (like poles).
• Oersted investigated the interaction between a magnet and the magnetic field of a conductor carrying current.
• We can ask questions about the forces between such current carrying wires and magnets that surround them.
• Consider a current carrying wire in a magnetic field
  • Several observations can be made
    • The force on the wire is perpendicular to Both the magnetic field and the current.
    • Reversing the current direction or field direction reverses the direction of the force.
    • Force is directly proportional to the magnetic field.
    • Force is directly proportional to the current.
    • Force is directly proportional to the length of the wire.
    • Force is directly proportional to the sine of the angle between the magnetic field and the current.
  • Combining the third to sixth observation leads to:

F a l I B sinq

or

F = K l I B sinq

• If the units of magnetic field are defined in terms of force, current and length, then K = 1, and we therefore can write.

F = l I B sinq

• This equation can now be written in its final form as

F = l I X B

• F = force, l = length, I = current and B = Magnetic field strength. Note that X indicates the cross product as described in section One Chapter 2.
• The direction of the force is determined by
  • Right Hand Rule Number 3 - Point your fingers on your right hand in the direction of the first vector listed in the formula (In this case I), twist your arm so that you can curl your fingers into the direction of the second vector listed in the formula (In this case B), the resulting vector will be in the direction that your thumb points (In this case F).

• The SI unit of magnetic field strength is the Tesla (T).
  • Tesla - is the magnetic field strength when a conductor with a current of one Ampere, and a length of 1 m at an angle of 90^o to the field, experiences a force of one Newton.

1 T = 1 N/A.m

• Make sure you review the rules for vector multiplication in section One Chapter 2.
• For F = lI X B,
  • F is a maximum if I is perpendicular to B
  • F is a minimum = 0 N if I is parallel to B

Forces on Moving Charges

• The cause of the force on a conductor carrying current placed in a magnetic field, is the movement of charges inside the conductor.
• It therefore makes sense that a free charge moving inside of a magnetic field should also experience a force.
• Consider n charges each of charge q moving inside a conductor of length l. These charges move length l in the time T.

• Force (F) is a maximum when v is perpendicular to B and is a minimum = 0 when v is parallel to B.
• The direction of the force is determined by right hand rule number 3
• If the charge q is negative, then the force F is negative and therefore the direction of the force is opposite to the direction that the thumb points in when using our right hand rule.
• These principles are often applied in particle accelerators, which are machines used to accelerate charged particles to very high speeds.
• The nature of the path taken by the particle in a magnetic field is
  • Case 1:
    • Velocity is perpendicular to the magnetic field - particle travels in a circular path

• In this first case, the centripetal force on the particle is given by the magnetic force.

F_mag = F_c or

qvB sinq = mv²/r or

r = (mv)/(qBsinq)

• Note that when v is perpendicular to B, q = 90^o and sinq = 1
• Case 2:
  • Velocity is parallel to the magnetic field - particle travels in a straight line.
    • The angle between the velocity and the field is zero, therefore, sin(0) = 0 and the force on the particle is zero. Zero force means, by Newton's first law of motion, that the particle goes straight.
• Case 3:
  • The velocity is neither parallel nor perpendicular to the magnetic field - The particle spirals along the magnetic lines.
    • There is a component of the veloctiy which is perpendicular to the magnetic field. This component makes the particle go into a circular path. There is also a component of the velocity that is parallel to the magnetic field. This parallel component carries the particle in the direction of the parallel component of velocity. Therefore, in the time it takes the particle to circle around once by the perpendicular component, the parallel component carries the particle sideways. The combination of these two motions then means the particle spirals along the magnetic lines.
    • The radius of the spiral is still found by the equation derived under case 1, only q is not equal to 90^o.
• Particle accelerators sometimes use these principle to accelerate the charged particles.
• Sometimes accelerators will accelerate particles through a potential difference to increase there speed.

Applications of Magnetic Forces on Conductors

• The principle that is applied is:
  • When a conductor is in a magnetic field, and the conductor has a current in it, then a force on the conductor occurs
  • The force on the conductor can cause motion.
  • We already saw two applications. They were the moving coil loud speaker and the galvanometer described in section one chapter 6.
  • Another application not discussed yet is the electric motor.

CONSIDER SECTlON YZ

• The right hand rule tells us the force on this section is up. This causes the coil to turn clockwise.
• Once the plane of the coil is vertical, the gaps in the commutator are over the brushes and no current flows.
• Inertia causes the coil to continue turning.
• The brushes are now on the opposite commutators. This causes the current in the coil to reverse. The current flows from z to y instead of from y to Z.
• The right band rule tells us that now section YZ (which is where WX is in the diagram), has a downwards force on it. This causes continued rotation of the loop.
• This process continues as the motor turns.

CONSIDER A DIFFERENT VIEW

Part I of the rotation

• Current flows in the armature as shown. If we grasp the armature with our right hand so that our finger tips point in the direction of conventional current, our thumb will point in the direction of the magnetic field inside of the coil of wires (towards the north pole of the electro magnet produced). Therefore the armature will have poles as shown. Like poles repel and therefore the armature turns.

Part II of the rotation

• The top brush is still in contact with commutator A and so the current is still in the same direction. The armature has rotated so that unlike poles attract. This continues the rotation.

Part III of the rotation

• The top brush is now contacting commutator B and so current has reversed direction. This causes the poles to reverse directions as shown (verify this with the rule outlined in Part I). Again, like poles repel and the rotation continues.

Part IV of the rotation

• The same comments can be made as in Part II, except the top brush is still in contact with commutator B.
  • Some definitions with respect to motors.
    • Commutator - A split Copper ring.
    • Brushes - pieces of carbon that rub on the commutator to pass current on to the coil.
    • Armature - a coil with a large number of windings around a soft iron core.
      • the core increases the field strength produced by the coil, and therefore increases the power of the motor.
    • Rheostat - a variable resistor

January 14, 2014