Magnetic flux and flux linkage

1. Magnetic flux:

• Magnetic flux density, which measures the strength of a magnetic field B, or B-field, in teslas (T).
• А diagram of a magnetic field indicates the density of magnetic flux by showing the number of flux lines per square metre (Figure 1).
• 
  Figure 1 Magnetic flux line passing through a surface area
• Magnetic flux is defined as magnetic flux density B (in teslas, T) multiplied by the area of the surface, A (in m²), where the area A is perpendicular to the lines of flux (Figure 1).
• Written as an equation, this becomes
• [math] \text{Flux} = B A [/math]
• Magnetic flux is measured in webers (Wb), where 1 Wb equals 1 T m².
• When an area is not perpendicular to the lines of magnetic flux, as shown in Figure 1, the flux through the area A is now the component.
• [math] \Phi = B A \cos \theta [/math]
• Magnetic flux has several applications in physics and engineering, including:
  1. Electromagnetic induction: A changing magnetic flux can induce an electromotive force (EMF) in a conductor.
  2. Magnetic circuits: Magnetic flux is used to analyze and design magnetic circuits, such as those found in motors, generators, and transformers.
  3. Particle physics: Magnetic flux is used to study the properties of subatomic particles, such as their magnetic moments.

2. Magnetic flux linkage:

• Magnetic flux linkage is defined as , where is the number of flux lines that pass through, or link, with each of the turns of a coil of N turns.
• Since
• [math] \text{flux} \, \Phi = BA [/math]
• for a single loop of wire, then the flux linkage is
• [math] N = BAN [/math]
• if the coil of wire has N turns that are perpendicular to the lines of flux.
  Flux linkage is measured in weber-turns.
• Flux linkage depends on several factors, as shown in Figure 2:

  Figure 2 Flux lines

  (a) the flux density
  (b) the orientation of the coil and flux lines
  (c) the coil’s cross-sectional area
  (d) the number of turns on the coil.

• Flux linkage is important because an e.m.f. is induced in a coil, in which the flux linkage changes. You will learn more about this in the next section.
• Figure 1 (c) shows a coil being turned in a magnetic field. As the coil turns in the magnetic field, the area of the coil perpendicular to the field is given by , and the magnetic flux linkage is given as
• [math]  N \Phi = BAN cos \theta [/math]
• where N is the number of turns on the coil, o is the magnetic flux (in Wb), B is the magnetic flux density (in T), A is the cross-sectional area of the coil (in m²) and is the angle between the axis of the coil and the flux lines.

• ⇒ Example:

• Figure 3 shows a coil of wire formed as a 60° triangle with sides of length 30 cm. The coil has 50 turns. Calculate the magnetic flux linkage with the coil when it is placed with the axis at 40° to a vertical in a uniform horizontal flux of 0.02 T.

  
  Figure 3

• Given data:
  Magnetic field = B = 0.02T
  Number of turns = N = 50
• Find data:
  Find the area of coil =?
  Magnetic flux linkage = NΦ =?
• Formula: 
• [math] \text{Area of coil} = \frac{1}{2} * \text{base} * \text{height} \\ N \Phi = B A \cos \theta [/math]
• Solution:
• Area of coil is
• [math] \text{Area of coil} = \frac{1}{2} * \text{base} * \text{height} \\ \text{Area of coil} = \frac{1}{2} \cdot 0.30\,\text{m} \cdot 0.30\,\text{m} \cdot \sin 60^\circ \\ \text{Area of coil} = 0.039\,\text{m}^2 [/math]
• Magnetic flux linkage is
• [math]N \Phi = B A N \cos \theta \\ N \Phi = (0.02)(0.039)(50) \cos 40^\circ \\ N \Phi = 0.03 \, \text{Wb} [/math]

Electromagnetic Induction

• Our lives would be completely different without Michael Faraday’s discovery of electromagnetic induction in 1831 using insulated coils of wire and changing magnetic fields.
• (1)

  
  Figure 4 using a moving magnet to induce an e.m.f.

• Electromagnetic induction using a coil of wire connected to a microammeter, as shown in Figure 4.
• The microammeter flicks one way when a bar magnet is moved into the coil, and the other way when the magnet is pulled out. It is zero when the magnet is stationary inside the coil.
• An e.m.f. is induced if there is relative movement between the coil and a magnetic field (either the magnet or the coil moves) or the magnetic flux linkage changes (for example, the strength of an electromagnet changes).
• (2)
• Figure 8.10 shows electromagnetic induction caused by a length of wire moving between two magnets.

  
  Figure 5 Moving a wire into a magnetic field induces an e.m.f.

• The wire is connected to the microammeter, which flicks one way when the wire moves down, and flicks in the opposite direction when the wire moves up.
• An e.m.f. is induced in the wire because an electric charge moving perpendicular to a magnetic field experiences a force.
• Using Fleming’s left-hand rule, you can see that electrons in a wire move towards one end of the wire when the wire moves perpendicular to the magnetic field.
• This leaves one end of the wire negatively charged overall and the other end positively charged, creating a potential difference across the wire.
• A current can flow if the wire is part of a complete circuit for example, when the wire is connected to a microammeter.

3. Faraday’s law and Lenz’s law:

• ⇒ Faraday’s law

• Faraday’s Law of Induction is a fundamental principle in electromagnetism, describing the relationship between a changing magnetic flux and the electromotive force (EMF) it produces. It states:
• “The electromotive force (EMF) induced in a closed loop is proportional to the rate of change of the magnetic flux through the loop.”

  
  Figure 6 Flux linkage in the coil increases as the coil moves closer to the magnet.

• [math] \varepsilon = \frac{\Delta (N \Phi)}{\Delta t} [/math]
• Where Δ(NΦ) is the change in flux linkage and Δt is the time over which that change takes place. Since a coil of wire has a fixed number of turns, this becomes
• [math] \varepsilon = N \frac{\Delta \Phi}{\Delta t} [/math]
• Some earlier observations:
• • Relative movement between a magnet and a coil changes the flux linkage in the coil (Figure 6). This generates an e.m.f.
• • Rotating a coil in the plane perpendicular to the field changes the cross- sectional area through which the flux passes. This changes the flux linkage, and generates an e.m.f.
• • Increasing the relative motion, or the speed at which the coil rotates, increases the rate of change of the flux linkage, which increases the induced e.m.f.
• • If there is no relative movement or rotation, the flux linkage does not change, so no e.m.f. is generated.

• ⇒ Lenz’s law:

• Lenz’s Law is a fundamental principle in electromagnetism that describes the direction of the induced current in a conductor when the magnetic flux through the conductor changes. The law states:
  “The induced current flows in a direction such that the magnetic field it produces opposes the change in magnetic flux that induced the current.”
• In other words, Lenz’s Law indicates that the induced current will always try to maintain the original magnetic flux by generating a magnetic field that:
  – Opposes an increasing magnetic flux (flux is increasing → current flows to decrease flux)
  – Supports a decreasing magnetic flux (flux is decreasing → current flows to increase flux)
• Lenz’s Law is often remembered using the following mnemonic:
  “Backward current opposes the change, forward current supports the change”
• Figure 7 shows the south pole of a magnet moving into a coil. This induces an e.m.f.

  
  Figure 7 A magnet being pushed into, or pulled out of, a coil of wire. Lenz’s law determines the direction of the induces e.m.f

• When there is a complete circuit, a current-flows and the coil behaves as an electromagnet, with its south pole facing the magnet’s south pole, repelling the magnet.
• Pulling the magnet out of the coil induces an e.m.f. such that the same end of the coil becomes a north pole, which attracts the magnet.
• We can combine Lenz’s law with Faraday’s law and write
• [math] \varepsilon = – \frac{\Delta (N \Phi)}{\Delta t}[/math]
• where Δ(NΦ)  is the change in flux linkage and Δt is the time over which that change takes place. Since a coil of wire has a fixed number of turns, this becomes
• [math] \varepsilon = – N\frac{\Delta \Phi}{\Delta t}[/math]
• Lenz’s law is the result of conservation of energy.
• When the south pole of a magnet is pushed into the coil, a current is induced in the wire, which becomes an electromagnet.
• If the south pole of the electromagnet laces the moving magnet, the poles repel and work must be done to keep pushing the magnet into the coil of wire.
• If you try this with a very strong magnet in a large coil, you may feel the force you are working against.

4. Application of magnetic field:

• ⇒ Conductor moving in a straight line:

• An induced e.m.f. can be caused by a conductor moving in a magnetic field.
• For example, a straight wire may be dropped through a uniform magnetic field, or a plane may fly at a constant height and speed in the Earth’s magnetic field.
• A credit card includes information stored on a magnetic strip.
• The credit card reader has a small coil in it, and when the credit card is swiped through the reader, an e.m.f. is induced in the coil.
• It is important to swipe the card quickly enough so that the induced e.m.f. is large enough to be interpreted.
• When a conductor moves at a velocity v perpendicular to the flux lines, Faraday’s law applies and an e.m.f. is generated.
• For a conductor of length, (l) travelling in a flux density B, the area swept out per second is length* velocity. The induced e.m.f. equals the rate of change of flux linkage, so
• [math] \varepsilon = B \frac{dA}{dt} [/math]
• Because the area swept out per second is , this becomes
• [math] \varepsilon  = Blv [/math]
• Where B is the magnetic flux density (in T), l is the length of the conductor (in m) and v is the velocity of the conductor perpendicular to the field (in ms¹).

5. Calculating an induced e.m.f. for a rotating coil:

• When a coil rotates in a magnetic field (figure 8), an a.c. voltage is induced in the coil.

  
  Figure 8 Remember to specify the axis of rotation for a coil in a magnetic field, which is shown here as the black dot.

• To calculate the value of the induces e.m.f. at time (t) , the following equation for a plane coil in a uniform magnetic field so long as the axis of rotation is at right angle to the field:
• [math] \varepsilon = BAN\omega \sin \omega t [/math]
• Where ε is the induced e.m.f. (in V), B is the magnetic flux density (in T), A is the cross-sectional are a of the coil (in m²), N is the number of turns on the coil, ω is the angular speed of the rotating coil (which can also be expressed as ω = 2πf, where f is the frequency of rotation current) and t is the time (in s).
• Since the maximum value of sint is 1, the maximum induced e.m.f. is
• [math] \varepsilon_{\text{max}} = BAN\omega [/math]
• Figure 9 shows how the magnetic flux linkage and the induced e.m.f. are linked:
• 
  Figure 9

• At A, NΦ is a maximum, gradient [math] \frac{d(N\Phi)}{dt}[/math] is 0, ε = 0.
• At B, NΦ is 0, gradient [math] \frac{d(N\Phi)}{dt}[/math] is maximum and negative, so ε is a maximum positive.
• At C, NΦ is a minimum, gradient  [math] \frac{d(N\Phi)}{dt}[/math] is 0, so ε = 0.
• At D, NΦ is 0, gradient [math] \frac{d(N\Phi)}{dt}[/math] is a maximum and positive, so ε is a minimum negative.