Electromagnetism

 Module 6: Field and particle physics

6.1 Fields
6.1.1

Electromagnetism

a)   Describe and explain:

I)                   The action of a transformer: magnetic flux from a coil; induced e.m.f = rate of change of flux linkage

II)                The action of a dynamo: change of flux linked produced by relative motion of flux and conductor

III)             Electromagnetic forces; qualitatively as arising from tendency of flux lines to contract or interaction of induced poles; quantitative calculation limited to force on a straight current-carrying wire in a uniform field

IV)             Simple linked electric and magnetic circuits: flux produced by current turns, need for large conductance and permeance and the effect of increasing the dimensions of an electromagnetic machine; qualitative effect of iron and air gap

b)   Make appropriate use of:

I)                   The terms: B-field, magnetic field, flux, flux linkage, induced e.m.f, eddy currents

by sketching and interpreting:

II)                Graphs of variations of currents, flux and induced e.m.f

III)             Diagrams of lines of flux in magnetic circuits; continuity of lines of flux

c)    Make calculations and estimates involving:

I)            [math]∅ = BA, ε = – \frac{d(∅N)}{dt}[/math]

II)          [math]F = ILB[/math]

III)        [math]\frac{V_1}{V_2} = \frac{N_1}{N_2}[/math]

IV)          [math]\frac{I_1}{I_2} = \frac{N_1}{N_2}[/math]    for an ideal transformer

d)   Demonstrate and apply knowledge and understanding of the following practical activities (HSW4):

I)                   Observing induced e.m.fs produced under varying conditions such as dropping a magnet through a coil attached to a data logger or oscilloscope

II)                Determining the uniform magnetic flux density between the poles of a magnet using a rigid current carrier and digital balance.

III)             Investigate transformers.
  • a) Describe and explain:

  • I) The Action of a Transformer

  • ⇒ Principle of Operation
  • A transformer transfers electrical energy from one circuit to another via electromagnetic induction. Its operation is based on Faraday’s Law of Electromagnetic Induction, which states:
  • [math]\text{Induced e.m.f} = -\frac{d\Phi}{dt}[/math]
  • Where:
  • – Φ is the magnetic flux.
  • – [math]\frac{d\Phi}{dt}[/math]​ is the rate of change of magnetic flux.
  • Figure 1 Electromagnetic Induction
  • Key Components:
  • – Primary Coil: Carries the input alternating current (AC), generating a time-varying magnetic field.
  • – Magnetic Core: Guides the magnetic flux and minimizes losses.
  • – Secondary Coil: Experiences a changing flux linkage, inducing an output e.m.f.
  • ⇒ Flux Linkage and Induced e.m.f.:
  • The flux linkage NΦ (where N is the number of turns) changes over time, and the induced e.m.f. in the secondary coil is given by:
  •  [math]\text{e.m.f in secondary coil} = -N_s \frac{d\Phi}{dt} [/math]
  • – For an ideal transformer:
  • [math]\frac{V_p}{V_s} = \frac{N_p}{N_s}[/math]
  • Where:
  • – [math]V_p[/math]: Primary voltage.
  • – [math]V_s[/math]: Secondary voltage.
  • – [math]N_p[/math]: Turns in the primary coil.
  • – [math]N_s[/math]​: Turns in the secondary coil.
  • ⇒  Applications:
  • – Step-up Transformer: Increases voltage for power transmission.
  • Figure 2 Set-up transformer
  • – Step-down Transformer: Decreases voltage for safe usage.
  • Figure 3 Step down transformer

  • II) The Action of a Dynamo

  • A dynamo converts mechanical energy into electrical energy via electromagnetic induction. It works on Faraday’s Law, using the relative motion between a conductor and a magnetic field to induce an e.m.f.
  • ⇒ Mechanism:
  • 1. Magnetic Flux:
  • – A magnetic field is established using permanent magnets or electromagnets.
  • 2. Relative Motion:
  • – A conductor (like a rotating coil) moves through the magnetic field.
  • 3. Induced e.m.f.:
  • – The changing flux linkage due to motion induces an e.m.f. in the coil.
  • ⇒  Equation:
  • The induced e.m.f. is proportional to the rate of change of flux linkage:
  • [math]\text{Induced e.m.f} = -\frac{d\Phi}{dt}[/math]
  • – For a coil with NNN turns:
  • [math]\text{Induced e.m.f} = -N \frac{d\Phi}{dt}[/math]
  • ⇒ Factors Affecting Output:
  • – Speed of Rotation: Faster rotation increases the rate of flux change and thus the e.m.f.
  • – Magnetic Flux Strength: Stronger fields induce higher e.m.f
  • – Number of Turns: More turns result in greater flux linkage.
  • ⇒  Applications:
  • – Dynamos in bicycles generate electrical energy to power lights.
  • – Larger dynamos are used in power stations for electricity generation.

  • III) Electromagnetic Forces

  • Electromagnetic forces arise from the interaction between current-carrying conductors and magnetic fields. These forces can be explained qualitatively and quantitatively:
  • Figure 4 A magnet produce electromagnetic force
  • ⇒ Qualitative Description:
  • 1. Flux Lines Contract:
  • – Magnetic field lines have a natural tendency to contract, creating forces.
  • 2. Induced Poles Interact:
  • – The poles created by induced currents interact with external fields, producing forces.
  • ⇒  Quantitative Description:
  • The force on a straight current-carrying wire in a uniform magnetic field is given by:
  • [math]F = BILsin θ[/math]
  • Where:
  • – F: Magnetic force (N).
  • – B: Magnetic flux density (T).
  • – I: Current in the wire (A).
  • – L: Length of the wire in the field (m).
  • – θ: Angle between the wire and the magnetic field.
  • ⇒   Special Cases:
  • Wire Parallel to Field ([math]θ = 0^0[/math]):
  • [math]F = 0 (\text{No force is experienced})[/math]
  • Wire Perpendicular to Field ([math]θ = 90^0[/math]):
  • [math]F = BIL [/math]
  • ⇒ Applications:
  • Electric motors rely on electromagnetic forces to produce rotational motion.
  • Loudspeakers use these forces to vibrate diaphragms and produce sound.

  • IV) Simple Linked Electric and Magnetic Circuits

  • ⇒ Magnetic Circuits:
  • Analogous to electric circuits, magnetic circuits are formed when magnetic flux flows through a path.
  • ⇒  Parameters:
  • Flux (Φ):
  • – Total magnetic field passing through a surface.
  • Magnetomotive Force (MMF):
  • [math]MMF = NI[/math]
  • – Where N is the number of turns and I is the current.
  • Reluctance ([math]R[/math]):
  • – Resistance to magnetic flux, analogous to electrical resistance:
  • [math]R = \frac{1}{μA}[/math]
  • – Where l is the length, μ is the permeability, and A is the cross-sectional area.
  • ⇒  Effect of Permeability:
  • Iron Core:
  • – High permeability (μ) concentrates flux, improving efficiency.
  • Air Gap:
  • – Low permeability increases reluctance, reducing flux.
  • ⇒  Practical Considerations:
  • Increasing Dimensions:
  • – Larger cross-sectional area (A) reduces reluctance, increasing flux.
  • – Longer paths (l) increase reluctance, decreasing flux.
  • Conductance and Permeance:
  • – Magnetic conductance ([math]\lambda[/math]) is analogous to electrical conductance.
  • – Permeance:
  • [math]\lambda = \frac{l}{R} \\
    \lambda = \frac{\mu A}{l}[/math]
  • ⇒  Applications:
  • Electromagnetic Machines:
  • – Transformers use high-permeability cores to maximize flux linkage.
  • – Motors and generators require optimized dimensions for efficient flux transfer.
  • Magnetic Storage:
  • – Magnetic circuits are essential in devices like hard drives.
  • Reluctance and Air Gaps:
  • – Air gaps are used to control magnetic flux in certain devices, such as magnetic levitation systems.

  • b) Make appropriate use of:

  • I) Terms and Definitions

  • ⇒ B-field (Magnetic Field):
  • Represents the magnetic field in a region.
  • – Symbol: B;
  • – SI unit: Tesla (T).
  • – It quantifies the strength and direction of the magnetic influence.
  • Properties:
  • – Lines of B-field: Represent the direction and strength; closer lines mean stronger fields.
  • – Right-hand Rule: The direction of the magnetic field around a current-carrying wire can be determined using the right-hand rule.
  • ⇒ Magnetic Flux (Φ):
  • Measures the total magnetic field passing through a given area.
  • – Formula:
  • [math]Φ = B ⋅ A \\ Φ = BA cosθ[/math]
  • Where:
  • – B: Magnetic field strength (T).
  • – A: Area perpendicular to the field (m²).
  • – θ: Angle between B and the perpendicular to A.
  • Unit: Weber (Wb).
  • ⇒ Flux Linkage (NΦ):
  • The total flux linked with a coil of N turns
  • Formula:
  • [math] \text{Flux Linkage} = NΦ[/math]
  • – Plays a key role in electromagnetic induction.
  • ⇒ Induced e.m.f.:
  • Electromotive force (e.m.f.) induced in a conductor due to a change in magnetic flux
  • Based on Faraday’s Law:
  • [math]\text{Induced e.m.f} = -\frac{d\Phi}{dt}[/math]
  • – Lenz’s Law dictates the polarity of the induced e.m.f.: It opposes the change in flux.
  • ⇒ Eddy Currents:
  • Loops of circulating currents induced within a conductor when exposed to a changing magnetic field.
  • Causes:
  • – Eddy currents arise due to Faraday’s Law in bulk conductors.
  • Effects:
  • – Heating: Energy loss as heat.
  • – Opposition to Motion: Generates forces opposing the source of induction.
  • Reduction:
  • – Laminated cores or segmented conductors are used to minimize eddy currents in devices like transformers.

  • II) Graphs of Variations of Currents, Flux, and Induced e.m.f.

  • ⇒ Current vs Flux:
  • The relationship between current (I) and magnetic flux (Φ) is fundamental in electromagnetism. According to Faraday’s Law of Induction, the induced electromotive force (e.m.f) in a circuit is proportional to the rate of change of magnetic flux.
  • The relationship between current and flux depends on whether the system is time-varying or not.
  • – If current is constant, the magnetic flux is also constant.
  • – If current is time-varying, the flux linked to the current varies too.
  • ⇒ Graph of Current (I) vs Flux (Φ):
  • Constant current (DC):
  • – If current remains constant, the magnetic flux also remains constant.
  • Varying current (AC):
  • – If current is alternating, the flux will also fluctuate sinusoidally, following the pattern of the current.
  • ⇒ Interpretation:
  • When current increases in a circuit with an inductor, the flux increases proportionally. If current decreases, the flux decreases.
  • For an alternating current, the magnetic flux will continuously reverse in direction, matching the frequency and shape of the AC.
  • ⇒ Induced e.m.f. vs Flux:
  • Faraday’s Law of Induction states that the induced e.m.f (ε) is equal to the negative rate of change of magnetic flux, or:
  • [math]\mathcal{E} = -\frac{d\Phi}{dt}[/math]
  • – As the magnetic flux changes over time, an induced e.m.f is produced in the circuit.
  • ⇒ Graph of Induced e.m.f. (ε) vs Flux (Φ):
  • When flux is increasing/decreasing linearly (uniform rate):
  • – The induced e.m.f will be constant (but with a sign opposite to the change in flux as per Lenz’s Law).
  • When flux changes non-linearly (for example sinusoidal AC):
  • – The induced e.m.f. will be a sinusoidal wave with a phase difference.
  • ⇒ Interpretation:
  • A rapidly changing flux will induce a high e.m.f, while a slow change will induce a low e.m.f. The induced e.m.f is always in the opposite direction of the change in flux, as per Lenz’s law.

  • III) Diagrams of Lines of Flux in Magnetic Circuits; Continuity of Lines of Flux

  • ⇒ Magnetic Flux and Flux Lines:
  • Magnetic flux is the measure of the number of magnetic field lines passing through a given area.
  • The magnetic field lines are visualized as lines that represent the direction and strength of the magnetic field. The density of these lines represents the field strength: the closer the lines, the stronger the field.
  • Key Points:
  • – The flux lines always form closed loops. In a magnet, they emerge from the north pole and curve around to the south pole.
  • – Inside the magnet, the lines continue from the south pole back to the north pole.
  • – The magnetic flux density (B) is proportional to the number of flux lines per unit area.
  • ⇒ Diagrams of Magnetic Flux Lines in Magnetic Circuits:
  • – A magnetic circuit consists of a source (e.g., a magnet or an electromagnet) and a closed loop path for the magnetic flux.
  • – Magnetic flux lines flow from the north to south pole of a magnet or from one core of a solenoid to the other, following the path of least resistance (magnetic reluctance).
  • – The flux lines prefer high permeability materials and tend to avoid materials with low permeability (such as air).
  • Example Diagrams:
  • – In the case of an electromagnet, the flux lines are concentrated within the core (usually made of iron or steel), and the lines spread out as they move away from the core.
  • – In a simple magnetic circuit with a magnetic coil (solenoid) and a core, the lines will form a continuous loop through the magnetic material, following the path defined by the core.
  • Figure 5 Magnetic circuit
  • ⇒  Continuity of Magnetic Flux Lines:
  • Continuity of magnetic flux lines means that magnetic flux cannot just “disappear.” Flux lines must always form a closed loop.
  • This principle is similar to the conservation of energy. In a magnetic circuit, the flux lines need to maintain continuity: if they enter a material, they must exit it as well, forming a loop.
  • The flux lines must continue in the material to the point where they can complete the loop, and this principle is crucial in the design of magnetic circuits (such as transformers or inductors), where the magnetic flux must always follow a continuous path through the core material.
  • ⇒ Graph of Current vs Flux:
  • Constant Current:
  • – Graph: A straight horizontal line indicating constant flux.
  • Alternating Current:
  • – Graph: A sinusoidal curve, showing a fluctuating magnetic flux, in sync with the alternating current.
  • ⇒  Graph of Induced e.m.f. vs Flux:
  • When Flux Changes:
  • – Graph: A sinusoidal curve showing e.m.f. generated as the flux changes.
  • ⇒ Magnetic Flux Lines Diagram (Magnetic Circuit):
  • A diagram of a magnetic circuit with a magnetic source (e.g., electromagnet) where lines of flux are visualized inside the core material.
  • Flux lines forming a continuous loop from the north pole to the south pole of the magnet and then through the core.
  • Figure 6 Graph between current and magnetic field

  • c) Make calculations and estimates involving:

  • I)  Magnetic Flux:

  • The equation for magnetic flux Φ is:
  • [math]Φ = B × A[/math]
  • Where:
  • – B is the magnetic flux density (in Tesla, T).
  • – A is the area through which the flux lines pass (in square meters, m²).
  • Calculation: If we know the magnetic flux density and the area, we can calculate the flux.
  • For example, let’s assume we have:
  • B=0.5 T
  • A=0.1 m2
  • – Then, the magnetic flux is:
  • [math]Φ = B × A \\ Φ = (0.1)(0.5) \\ Φ = 0.05 \text{Weber (Wb)}[/math]
  • So, the magnetic flux Φ is 0.05 Wb.

  • II)  Magnetic Force on a Current-Carrying Wire :

  • ⇒ Explanation:
  • When a current-carrying wire is placed in a magnetic field, it experiences a force due to the interaction of the magnetic field and the moving charges.
  • Formula:
  • [math]F = ILBsinθ[/math]
  •   Where:
  • – F: Magnetic force (N).
  • – I: Current in the wire (A)
  • – L: Length of the wire in the field (m).
  • – B: Magnetic flux density (T).
  • – θ: Angle between the wire and the magnetic field.
  • ⇒ Example Calculation:
  • A wire of length 5 m carries a current of 2 A and is placed in a uniform magnetic field of 0.1 T at an angle of [math]90^ o[/math].
  • Force:
  • [math]F = ILB \, sinθ \\ F = 2(0.5)(0.1) sin90^0 \\ F = 0.1 N[/math]
  • Result: The force on the wire is 0.1 N.

  • III) Transformer Equations

  • ⇒ Voltage Relationship:
  • In an ideal transformer, the voltage ratio is proportional to the turn’s ratio:
  • [math]\frac{V_1}{V_2} = \frac{N_1}{N_2} [/math]
  • Where:
  • ​ – [math]V_1[/math]: Voltage in the primary coil.
  • ​ – [math]V_2[/math]: Voltage in the secondary coil.
  •  – [math]N_1[/math]​: Number of turns in the primary coil.
  • ​ – [math]N_2[/math]: Number of turns in the secondary coil.
  • Figure 7 Working of transformer
  • ⇒ Example Calculation:
  • A transformer has [math]N_1[/math]​=100 turns in the primary coil and [math]N_2[/math] ​=50 turns in the secondary coil. If the input voltage is 240 V, find the output voltage.
  • 1. Voltage ratio:
  • [math]\frac{V_1}{V_2} = \frac{N_1}{N_2} [/math]
  • 2. Output voltage:
  • [math]V_2 = \frac{V_1 N_2}{N_1} \\
    V_2 = \frac{(240)(50)}{100} \\
    V_2 = 120V [/math]
  • Result: The output voltage is [math]120V[/math].
  • ⇒  Current Relationship:
  • In an ideal transformer, the current ratio is inversely proportional to the turns ratio:
  • [math]\frac{I_1}{I_2} = \frac{N_1}{N_2} [/math]
  • Where:
  • -​ [math]I_1[/math]: Current in the primary coil.
  • – [math]I_2[/math]: Current in the secondary coil.
  • ⇒  Example Calculation:
  • Using the same transformer as above ([math]N_1 = 100, N_2 = 50[/math]), if the output current is 4 A, calculate the input current.
  • – Current ratio:
  • [math]\frac{I_1}{I_2} = \frac{N_1}{N_2} [/math]
  • – Input current:
  • [math]I_1 = I_2 \frac{N_1}{N_2} \\
    I_1 = \frac{(4)(100)}{50} \\
    I_1 = 8A [/math]
  • Result: The input current is 8 A8A.

  • IV)  Combined Power Conservation

  • For an ideal transformer, power is conserved:
  • [math]P_1 = P_2[/math]
  •   Or
  • [math]V_1 I_1 = V_2 I_2 [/math]
  • ⇒  Example Check:
  • Using the transformer from above:
  • – Input power:
  • [math]P_1 = V_1 I_1 \\
    P_1 = (240)(8) \\
    P_1 = 1920 W
    [/math]
  • – Output power:
  • [math]P_2 = V_2 I_2 \\
    P_2 = (120)(16) \\
    P_2 = 1920 W[/math]
  • Result: Power is conserve:

  • d) Demonstrate and apply knowledge and understanding of the following practical activities (HSW4):

  • I)  Observing Induced e.m.f.s: Magnet and Coil Experiment

  • ⇒ Objective:
  • To observe the variation in induced e.m.f. as a magnet moves through a coil under different conditions and study the factors affecting it.
  • ⇒  Apparatus:
  • – Strong bar magnet.
  • – Coil of wire connected to a data logger or oscilloscope.
  • – Stand to hold the coil.
  • – Meter ruler.
  • Figure 8 Induced e.m.f from magnet and coil
  • ⇒  Procedure:
    • Set up the coil vertically and connect it to a data logger or oscilloscope.
    • Drop a bar magnet through the coil from a fixed height. Ensure the poles of the magnet align with the coil axis.
    • Observe and record the induced e.m.f. as the magnet moves through the coil.
    • Repeat the experiment under varying conditions:
  • – Change the speed of the magnet (drop from different heights).
  • – Use magnets of different strengths (vary B)
  • – Vary the number of turns in the coil (N).
  • ⇒ Observations:
  • The oscilloscope will show a voltage-time graph with two peaks:
  • – A positive peak as the magnet enters the coil.
  • – A negative peak as the magnet leaves the coil.
  • The induced e.m.f. depends on:
  • – The speed of the magnet (​[math]\frac{d\Phi}{dt}[/math])
  • – The number of turns in the coil (N)
  • – The strength of the magnet (B).
  • Faraday’s Law:
  • [math] \varepsilon = -\frac{d(N\Phi)}{dt}[/math]
  • – Faster movement and stronger magnets induce larger e.m.f.s.
  • – Opposite peaks occur due to the changing direction of flux linkage.

  • II)  Determining Magnetic Flux Density Using a Rigid Current Carrier

  • ⇒  Objective:
  • To measure the uniform magnetic flux density (BBB) between the poles of a magnet using a current-carrying wire and a digital balance.
  • ⇒  Apparatus:
  • – Rigid current-carrying wire mounted on a holder.
  • – Magnet with uniform field between its poles.
  • – Digital balance.
  • – Power supply and ammeter.
  • – Measuring calipers for length of the wire in the field.
  • Figure 9 Measuring magnetic flux density
  • ⇒  Procedure:
    • Place the wire horizontally between the poles of the magnet so it is perpendicular to the field.
    • Set up the circuit to allow current to flow through the wire.
    • Measure the initial mass reading on the balance.
    • Switch on the power supply and note the new mass reading on the balance.
    • Record the current flowing through the wire using the ammeter.
    • Repeat the experiment for different currents.
  • ⇒  Calculation:
  • The force on the wire causes an apparent change in weight on the balance:
  • [math]F = ∆m . g[/math]
  • Where:
  • – Δm: Change in mass (kg).
  • – g: Acceleration due to gravity ([math]9.81 m/s^2[/math]).
  • Using
  • [math]F = ILB[/math]
  • Calculate B:
  • [math]B = \frac{F}{IL}[/math]
  • Where:
  • – F: Magnetic force (N).
  • – I: Current through the wire (A).
  • – L: Length of wire in the field (m).
  • ⇒  Observations:
    • The balance reading increases or decreases depending on the direction of the current.
    • The magnetic flux density can be calculated and verified for uniformity.

  • III. Investigating Transformers

  • ⇒  Objective:
  • To investigate the principles of transformers, including the relationship between voltage, current, and the number of turns in the primary and secondary coils.
  • ⇒  Apparatus:
  • – Transformer kit with adjustable coils.
  • – Power supply (AC source).
  • – Voltmeter and ammeter.
  • – Load resistor or lamp.
  • – Connecting wires.
  • ⇒   Procedure:
    • Set up the transformer with a known number of turns (​[math]N_1[/math]) in the primary coil and ([math]N_2[/math]) in the secondary coil.
    • Connect the primary coil to the AC power supply and the secondary coil to a load resistor.
    • Measure and record the primary voltage ([math]V_1[/math]) and current (​[math]V_2[/math]) using meters.
    • Measure and record the secondary voltage (​[math]V_2[/math]) and current ([math]I_2[/math]​).
    • Repeat the experiment for different [math]N_2[/math] values (secondary turns).
  • Figure 10 Investigate transformer
  • ⇒  Observations:
  • 1. Voltage Ratio:
  • [math]\frac{V_1}{V_2} = \frac{N_1}{N_2}[/math]
  • Verify that the voltage ratio matches the turns ratio.
  • 2. Current Ratio:
  • [math]\frac{I_1}{I_2} = \frac{N_1}{N_2}[/math]
  • Check that the current ratio is inversely proportional to the turns ratio.
  • 3. Power Conservation (for an ideal transformer):
  • [math]P_1 = P_2[/math]
  •      Or
  • [math]V_1 I_1 = V_2 I_2[/math]
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