Concept of Magnetic Fields

• A magnetic field is a region in which a magnet, a wire carrying current or a moving charge experiences a force
• It can be produced by:
  • a current carrying conductor
  • permanent magnets

Representing Magnetic Fields

• Magnetic Field due to a permanent magnet
  
  • Magnetic field lines come out of the North pole and go into the South pole
  • The direction of a field line at any point in the field shows the direction of the force that a ‘free’ magnetic north pole would experience
  • The field is strongest where field lines are closest to one another

Electromagnetism

• Strength of magnetic field can be increased by:
  • Increasing the current
  • Increasing the number of turns per unit length of the solenoid
  • Using soft-iron core within the solenoid
• Right-hand Grip Rule:
  
• For a long, straight conductor:
  • Magnetic field lines are concentric circles centered at the conductor
  • Separation between adjacent field lines increases with distance from the conductor
  • The magnetic field is non-uniform
    
• For a flat, circular coil:
  • The magnetic field pattern produced represents that produced by a short bar magnet
    
• For a solenoid and flat circular coil:
  • The magnetic field pattern produced is identical to that produced by a bar magnet
  • The magnetic field lines within the solenoid are parallel, indicating the strength is the same (uniform field)
    

Determining the Pole of a Magnetic Field

• It is determined by the Right Hand Grip rule, however, this time, the fingers represent the current
  

Effect of a Ferrous in a Solenoid

• The strength of the generated magnetic field can be increased (by about 1000 times) by adding a ferrous (iron) core inside the solenoid
• Two possible reasons to explain this effect:
  • Ferrous materials have a higher permeability than air; a stronger the ability to support the formation of a magnetic field within itself
  • Ferrous materials are magnetic and become magnetized when placed into the solenoid, thus contributing to the overall magnetic field strength of the solenoid

Force on a Current Carrying Conductor

• Fleming’s Left Hand Rule
  
• Force acting on a current carrying conductor in a magnetic field
  
• Strength of force can be increased by:
  • increasing the current
  • using a stronger magnet

Forces between Currents

• It can be worked our by considering one wire’s magnetic field (using the Right-Hand Grip rule), drawing a tangent at the position of the other wire and then applying Fleming’s Left-Hand rule

Magnetic Flux Density

• Magnetic Flux (Φ) is the number of magnetic field lines passing normally to a given area. Its unit is Weber (Wb)
• Magnetic Flux Density (B) is the force acting per unit current on a unit length of a conductor placed at right angles to the magnetic field
  • Φ = BA
• 1 Tesla is the magnetic field producing a force of 1 Nm^-1 on a wire carrying current of 1 A normal to the field
  • 1 T = 1 NA^-1m^-1
• The magnitude of the force on a current carrying conductor with:
  • F = BIL sin θ
  • Find the direction using Fleming’s Left-Hand rule
  • If the wire is parallel to the field lines, θ = 0 and F = 0
  • If the wire is at right angles to the field lines, θ = 90 and the force acting on the wire is maximum; F = BIL

Examples

1. Two long straight vertical wires X and Y pass through a horizontal card, carrying current upward. The magnetic flux density B at a distance x from a long straight wire due to a current I in the wire is given by B = ^μ₀I/_2πx
   The current in the wire X is 5.0 A and that in wire Y is 7.0 A. The separation of the wires is 2.5 cm
   1. Calculate the force per unit length on wire Y due to the current in wire X
      • Using the given expression, find B due to wire X by substituting the current in X and separate:
        • B = ^{4π × 10-7 × 5} / _{2π × 2.5 × 10^-2}
        • ∴ B = 4 × 10^-5
      • To find the force per unit length, divide the expression for force by length and substitute the appropriate values:
        • F = BIl ÷ l
        • ^F/_l = 4 × 10^-5 × 7
        • ∴ ^F/_l = 2.8 × 10^-4
   2. The currents in the wires are not equal. State and explain whether the forces on the two wires are equal in magnitude
      • The force due to the magnetic field depends on the product of the currents in the two wires, hence both values would be equal. Also, Newton’s third law applies and the reaction force is equal but opposite

Measuring Flux Density

• The force on a current carrying conductor can be used to measure the flux density of a magnetic field using a current balance
  
• small weights = mg
• Force due to the current = BIl
• Assuming forces act at the same distance from the pivot, so no need to take moments.
• Equate forces:
  • mg = BIl
  • B = ^mg/_Il

Forces on a Moving Charge

• F = BIl
• I = ^Q/_t
• ∴ F = BQ^l/_t
• but v = ^l/_t
• ∴ F = BQv
• If the particle is moving at an angle θ to the magnetic field, the component of velocity perpendicular to the magnetic field id v sin θ

The Hall Effect

• The Hall effect is a mechanism in which magnetic and electric forces on a moving charged particle are balanced
• The probe is made of a semiconductor material as electrons travel faster in it than in metal; the greater effect
• A small current flows through the probe and a magnetic field is applied so the electrons are pushed sideways by the magnetic force, accumulating on one side, hence producing a small voltage, which is called Hall Voltage
• The greater the flux density, the greater the Hall voltage
• If the magnetic field direction is reversed, electrons are pushed to the opposite side and the Hall voltage is reversed

The Hall Voltage

• An electric field is set up in the probe as there is a difference in voltage between a distance d, so:
  • E = ^V_H/_d
• As a single electron travels with a drift velocity v, it experiences a force to the left due to the magnetic field Bqv, and a force to the right due to the electric field Eq
• Soon an equilibrium is reached, hence the forces are equated
  • Eq = Bqv
• Substitute for E
  • ^qV_H/_d = Bqv
• Current is related to mean drift velocity by
  • I = nAvq
  • where A = td is the cross sectional area, and
  • n is the number density of conducting particles
• Substitute for v and rearrange
  • ^qV_H/_d = ^BqI/_ntdq
  • V_H = ^BI/_ntq

Deflection of e^– through a B-field

• Circular motion
  • Circular path
  • E_k constant
  • BQv = ^mv²/_r so r = ^mv/_BQ
• Faster moving particles move in bigger circles, r ∝ v
• Heavier particles move in bigger circles, r ∝ m
• For stronger fields, particles move in smaller circles, r ∝ ¹/_B

Charge-to-mass Ratio

• The charge-to-mass ratio is know as the specific charge on the electron
• Determination of ^e/_me
  • Work done by electron is equivalent to E_k, it possesses
    • W = QV
    • E_k = ½mv²
  • Using the equation for an electron travelling in a circle in a magnetic field to eliminate v
    • v = ^rBe/_me
    • eV = ½m_e(^rBe/_me)²
    • ^e/_me = ^2V/_r²B²

Deflection of e^– through an E-field

• Determining the motion of the electron
  • s = ut + ½at² and initial velocity = 0 ms-1, ∴ y = ½at²
  • Finding an equation for acceleration:
    • F = qE, and
    • F = ma
    • As the particle moves horizontally at constant velocity and time is the same for the whole journey
      • x = vt
      • t = ^x/_v
      • ∴ y = (^eE/_2mv²) · x²
    • Hence, y ∝ x², therefore parabolic (projectile) motion
    • Gain in the y-component of velocity, ∴ E_k increases

Crossed-Fields

• Considering a setup where electric and magnetic fields are perpendicular to each other and act on a moving charge simultaneously
• In such case, a certain velocity exists where the fields exert equal and opposite forces
  • B-field = E-field
  • BQv = QE
  • v = ^E/_B

  

• If velocity is higher, F = BQv, hence the magnetic force is stronger and the effect of the force due to the electric field decreases
• If the velocity is lower, F = BQv, hence the magnetic force is weaker and the effect of the force due to the electric field increases

Force in Gravitational, Electric and Magnetic Fields

Examples

1. A small mass is placed in a field of force that is either electric or magnetic or gravitational.
   State the nature of the field of the force when the mass is
   1. charged and the force is opposite to the direction of the field
      • Electric field
   2. uncharged and the force is in the direction of the field
      • Gravitational field
   3. charged and there is a force only when the mass is moving
      • Magnetic field
   4. charged and there is no force on the mass when it is stationary or moving in a particular direction
      • Magnetic field