Emf, Magnetic Flux & Magnetic Flux Linkage (DP IB Physics: HL)

Electromagnetic Induction

• When a conducting wire moves through a magnetic field, a potential difference is created along the wire
  • If the wire is part of a closed circuit then an e.m.f is induced

• We can produce a current in a wire simply by moving a magnet near to it
  • Electrical energy is produced by the system since work is done on the wire by moving the magnet relative to the free electrons within it

• Therefore, electromagnetic induction is the term applied when an e.m.f. is induced in a closed circuit conductor due to it moving through a magnetic field
  • Examples are a flat coil or a solenoid

• Electromagnetic induction happens when a conductor cuts through magnetic field lines
  • The amount of e.m.f induced is determined by the magnetic flux and the area on which the magnetic field acts
• An example of this is moving conductors in a magnetic field
  • Similar to a coil or a solenoid, a straight conducting rod moving through a magnetic field will also have an e.m.f induced in it
  • This is only when the conductor moves perpendicular to the magnetic field lines where it cuts through the magnetic field lines

Conducting rod moving perpendicular to a magnetic field directed into the page

• THe induced e.m.f in the conductor, as it moves through the magnetic field, is:

ε = BLv

• Where:
  • ε = emf (B)
  • B = magnetic flux density (T)
  • L = length of the conductor in the field (m)
  • v = velocity of the conductor travelling through the field (m s^–1)

• This equation shows that the e.m.f induced increases if:
  • A longer conductor is in the field
  • The magnetic field strength is larger
  • The conductor cuts through the field lines faster
• Coiling a wire to form many loops, or turns, will also increase the size of the induced e.m.f.
• For a wire in a magnetic field with N turns, the induced e.m.f. will become

ε = BLvN

Magnetic Flux

• Magnetic flux is defined as:

The product of the magnetic flux density and the cross-sectional area perpendicular to the direction of the magnetic flux density

• Magnetic flux when the field and motion are at 90^o can be calculated using the simple equation:

• Where:
  • Φ = magnetic flux (Wb)
  • B = magnetic flux density(T)
  • A = cross-sectional area (m²)

Changing Angle 

• The flux is the total magnetic field that passes through a given area
  • It is a maximum when the magnetic field lines are perpendicular to the plane of the area
  • It is zero when the magnetic field lines are parallel to the plane of the area

• For a coil, the amount of magnetic flux varies as the coil rotates within the field

• In other words, magnetic flux is the number of magnetic field lines through a given area

• When the magnetic field lines are not completely perpendicular to the area A, then the component of magnetic flux density B is perpendicular to the area is taken
• The equation then becomes:

Φ = BA cos(θ)

• Where:
  • Φ = magnetic flux (Wb)
  • B = magnetic flux density (T)
  • A = cross-sectional area (m²)
  • θ = angle between magnetic field lines and the line perpendicular to the plane of the area (often called the normal line) (degrees)

• This means the magnetic flux is:
  • Maximum = BA when cos(θ) =1 therefore θ = 0^o. The magnetic field lines are perpendicular to the plane of the area
  • Minimum = 0 when cos(θ) = 0 therefore θ = 90^o. The magnetic fields lines are parallel to the plane of the area

Part (a)

Φ = BA

Step 3: Substitute in values

Φ = (1.8 × 10⁻⁵) × 0.292 = 5.256 × 10⁻⁶ = 5.3 × 10⁻⁶ Wb

Part (b)

• The magnetic flux will be at a minimum when the window is opened by 90° and a maximum when fully closed or opened to 180°
• This is shown by the graph:

Magnetic Flux Linkage

• More coils in a wire mean a larger e.m.f is induced
• The magnetic flux linkage is a quantity commonly used for solenoids which are made of N turns of wire
• The flux linkage is defined as:

The product of the magnetic flux and the number of turns of the coil

• It is calculated using the equation:

Magnetic flux linkage = ΦN = BAN

• Where:
  • Φ = magnetic flux (Wb)
  • N = number of turns of the coil
  • B = magnetic flux density (T)
  • A = cross-sectional area (m²)

   

• The flux linkage ΦN has the units of Weber turns (Wb turns)

• An e.m.f is induced in a circuit when the magnetic flux linkage changes with respect to time
• This means an e.m.f is induced when there is:
  • A changing magnetic flux density B
  • A changing cross-sectional area A
  • A change in angle θ

The magnetic flux through a rectangular coil increases as the angle between the field lines and plane decreases

• Magnetic flux linkage also changes with the rotation of the coil
  • It is at a maximum when the field lines are perpendicular to the plane of the area they are passing through
• Therefore, the component of the flux density which is perpendicular is equal to:

ΦN = BAN cos(θ)

• Where:
  • N = number of turns of the coil