Electromagnetic induction

1. Overview

Electromagnetic induction is the fundamental physical process where a change in the magnetic environment of a conductor induces an electromotive force (e.m.f.) across it. This principle dictates that electricity can be generated by moving a conductor through a magnetic field or by changing the magnetic field strength surrounding a stationary conductor. It serves as the theoretical foundation for modern electrical infrastructure, including AC generators, transformers, and induction braking systems. The core of this phenomenon lies in the interaction between magnetic fields and the charges within a conductor, governed by the laws of Faraday and Lenz, which ensure the conservation of energy during the conversion of mechanical work into electrical energy.

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Key Definitions

• Magnetic Flux ($\Phi$): The product of the magnetic flux density and the cross-sectional area perpendicular to the direction of the magnetic flux density. It is a scalar quantity representing the total "amount" of magnetic field passing through a surface.
• Magnetic Flux Linkage ($N\Phi$): The product of the magnetic flux passing through a coil and the number of turns $N$ in that coil. It represents the total flux interacting with the entire length of the wire in a solenoid or coil.
• Weber (Wb): The SI unit of magnetic flux. One Weber is defined as the magnetic flux which, linking a circuit of one turn, produces an e.m.f. of one volt when the flux is reduced to zero at a uniform rate in one second. ($1\text{ Wb} = 1\text{ V s} = 1\text{ T m}^2$).
• Faraday’s Law of Electromagnetic Induction: The magnitude of the induced e.m.f. is directly proportional to the rate of change of magnetic flux linkage.
• Lenz’s Law: The direction of the induced e.m.f. (and the resulting induced current) is such that it opposes the change in magnetic flux that produced it. This is a statement of the Principle of Conservation of Energy.

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Content

3.1 Magnetic Flux and Flux Linkage

Magnetic flux density ($B$) describes the strength of a field at a point (flux per unit area), whereas magnetic flux ($\Phi$) describes the total field passing through a specific area ($A$).

The Fundamental Equation: $$\mathbf{\Phi = BA}$$ This equation applies only when the magnetic field is perfectly normal (perpendicular) to the plane of the area.

Accounting for Angle: In many scenarios, the magnetic field lines are not perpendicular to the surface. We must use the component of the magnetic flux density that is normal to the area. $$\mathbf{\Phi = BA \cos \theta}$$ Where:

• $\Phi$ = Magnetic Flux (Wb)
• $B$ = Magnetic Flux Density (T)
• $A$ = Cross-sectional area (m²)
• $\theta$ = The angle between the magnetic field lines and the normal (a line at $90^\circ$) to the plane of the area.

Magnetic Flux Linkage in Coils: When a wire is wound into a coil of $N$ turns, the magnetic flux passes through each individual loop. The total flux "linked" to the circuit is the sum of the flux through every turn: $$\mathbf{\text{Flux Linkage} = N\Phi = BAN \cos \theta}$$ Note: Flux linkage is measured in Weber-turns (Wb), though the unit is often simplified to just Webers in many contexts.

3.2 Demonstrating Electromagnetic Induction

The following experiments are essential for understanding how e.m.f. is induced and the factors that govern its magnitude and direction.

Experiment 1: Relative Motion between a Magnet and a Solenoid

• Setup: A long solenoid is connected to a sensitive center-zero galvanometer (which can show current in both directions). A bar magnet is moved along the axis of the solenoid.
• Observations:
  1. Stationary Magnet: When the magnet is held still inside or outside the coil, the galvanometer reads zero. There is flux, but no change in flux.
  2. Insertion: Moving the North pole into the coil causes a momentary deflection (e.g., to the right).
  3. Withdrawal: Pulling the North pole out causes a deflection in the opposite direction (to the left).
  4. Speed: Moving the magnet faster results in a larger deflection.
  5. Polarity: Moving a South pole into the coil produces a deflection opposite to that of the North pole.
• Conclusion: An e.m.f. is only induced when there is relative motion, causing the magnetic flux linkage to change over time.

Experiment 2: Mutual Induction (The Two-Coil Experiment)

• Setup: A "Primary" coil is connected to a DC power supply and a switch. A "Secondary" coil is placed close to (or wound over) the primary and connected to a galvanometer. There is no electrical connection between the two coils.
• Observations:
  1. Switch Closing: At the moment the switch is closed, the current in the primary rises from zero, creating an increasing magnetic field. This changing field links with the secondary coil, causing a momentary deflection.
  2. Steady State: While the switch is closed and the current is constant, the galvanometer returns to zero.
  3. Switch Opening: When the switch is opened, the collapsing magnetic field induces an e.m.f. in the secondary in the opposite direction.
• Conclusion: It is the rate of change of the magnetic field, not the presence of the field itself, that induces an e.m.f.

Experiment 3: Factors Affecting the Magnitude of Induced e.m.f. By systematically varying the setup, we observe that the induced e.m.f. increases when:

• The speed of the relative motion increases (decreasing $\Delta t$).
• The strength of the magnetic field ($B$) increases.
• The number of turns ($N$) on the coil increases.
• The cross-sectional area ($A$) of the coil increases.

3.3 Faraday’s and Lenz’s Laws

These two laws are combined into a single mathematical expression: $$\mathbf{E = -\frac{d(N\Phi)}{dt}}$$ Or, for a constant rate of change: $$\mathbf{E = -N\frac{\Delta\Phi}{\Delta t}}$$

Faraday's Law (The Magnitude): The magnitude of $E$ is determined by how quickly the flux linkage $N\Phi$ changes. If you graph $N\Phi$ against time $t$, the gradient of the graph at any point is equal to the magnitude of the induced e.m.f.

Lenz's Law (The Direction and the Negative Sign): The negative sign in the equation represents Lenz's Law. It indicates that the induced e.m.f. acts in a direction that creates effects (like an induced current) to oppose the change that caused it.

Lenz’s Law and the Conservation of Energy: This is a frequent exam topic. If the induced e.m.f. did not oppose the change, but instead aided it:

1. A North pole approaching a coil would induce a South pole at the near end.
2. The South pole would attract the North pole, accelerating the magnet towards the coil.
3. This would increase the magnet's kinetic energy while simultaneously generating electrical energy in the coil.
4. Energy would be created from nothing, violating the Principle of Conservation of Energy.

• Correct Physics: As a North pole approaches, the coil induces a current that creates a North pole at the near end to repel the magnet. An external agent must do work to push the magnet against this repulsive force. This mechanical work is the source of the electrical energy generated in the circuit.

3.4 Worked Examples

Worked example 1 — Rotating Coil in a Uniform Field

A square coil of side length $5.0\text{ cm}$ consists of $200$ turns of wire. It is placed in a uniform magnetic field of flux density $0.15\text{ T}$. Initially, the plane of the coil is perpendicular to the magnetic field. The coil is then rotated through $90^\circ$ about an axis perpendicular to the field in a time of $0.20\text{ s}$. Calculate the average e.m.f. induced.

Solution:

1. Calculate Area ($A$): $A = \text{side}^2 = (0.050\text{ m})^2 = 2.5 \times 10^{-3}\text{ m}^2$
2. Initial Flux Linkage ($(N\Phi)_{initial}$): The plane is perpendicular to the field, so the normal is parallel to the field ($\theta = 0^\circ$). $(N\Phi)_{initial} = BAN \cos(0^\circ) = 0.15 \times (2.5 \times 10^{-3}) \times 200 = 0.075\text{ Wb}$
3. Final Flux Linkage ($(N\Phi)_{final}$): The coil is rotated $90^\circ$, so the plane is now parallel to the field (the normal is at $90^\circ$ to the field). $(N\Phi)_{final} = BAN \cos(90^\circ) = 0\text{ Wb}$
4. Calculate Change in Flux Linkage ($\Delta(N\Phi)$): $\Delta(N\Phi) = 0 - 0.075 = -0.075\text{ Wb}$
5. Apply Faraday’s Law: $E = \left| \frac{\Delta(N\Phi)}{\Delta t} \right| = \frac{0.075\text{ Wb}}{0.20\text{ s}}$ $E = 0.375\text{ V}$
6. Final Answer: $E = 0.38\text{ V}$ (to 2 s.f.)

Worked example 2 — Changing Magnetic Field in a Solenoid

A small "search coil" with $40$ turns and an area of $1.2\text{ cm}^2$ is placed inside a large solenoid. The magnetic flux density $B$ inside the solenoid is given by the equation $B = 0.050I$, where $I$ is the current. The current in the solenoid is reduced from $4.0\text{ A}$ to $1.0\text{ A}$ in a time interval of $15\text{ ms}$. Calculate the e.m.f. induced in the search coil.

Solution:

1. Convert Units: $A = 1.2\text{ cm}^2 = 1.2 \times (10^{-2}\text{ m})^2 = 1.2 \times 10^{-4}\text{ m}^2$ $\Delta t = 15\text{ ms} = 0.015\text{ s}$
2. Calculate Change in Flux Density ($\Delta B$): $B_{initial} = 0.050 \times 4.0 = 0.20\text{ T}$ $B_{final} = 0.050 \times 1.0 = 0.050\text{ T}$ $\Delta B = 0.050 - 0.20 = -0.15\text{ T}$
3. Calculate Change in Flux Linkage ($\Delta(N\Phi)$): Since $A$ and $N$ are constant: $\Delta(N\Phi) = NA(\Delta B) = 40 \times (1.2 \times 10^{-4}) \times (-0.15)$ $\Delta(N\Phi) = -7.2 \times 10^{-4}\text{ Wb}$
4. Calculate e.m.f. ($E$): $E = -\frac{\Delta(N\Phi)}{\Delta t} = -\frac{-7.2 \times 10^{-4}}{0.015}$ $E = 0.048\text{ V}$
5. Final Answer: $E = 48\text{ mV}$

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Key Equations

Quantity	Equation	Symbols	Data Sheet?
Magnetic Flux	$\mathbf{\Phi = BA \cos \theta}$	$B$: Flux Density (T), $A$: Area (m²), $\theta$: angle to normal	No
Flux Linkage	$\mathbf{N\Phi = BAN \cos \theta}$	$N$: Number of turns	No
Faraday's Law	$\mathbf{E = \frac{d(N\Phi)}{dt}}$	$E$: induced e.m.f. (V), $t$: time (s)	Yes
Combined Law	$\mathbf{E = -N\frac{\Delta\Phi}{\Delta t}}$	Negative sign represents Lenz's Law	Yes

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Common Mistakes to Avoid

• ❌ Wrong: Using the angle between the field and the surface of the coil in $\Phi = BA \cos \theta$.
  • ✅ Right: $\theta$ is the angle between the field and the normal to the surface. If the field is parallel to the surface, $\theta = 90^\circ$ and $\Phi = 0$. If the field is perpendicular to the surface, $\theta = 0^\circ$ and $\Phi = BA$.
• ❌ Wrong: Forgetting to convert area from $\text{cm}^2$ or $\text{mm}^2$ to $\text{m}^2$.
  • ✅ Right: Remember that $1\text{ cm}^2 = 10^{-4}\text{ m}^2$ and $1\text{ mm}^2 = 10^{-6}\text{ m}^2$.
• ❌ Wrong: Confusing Magnetic Flux ($\Phi$) with Magnetic Flux Density ($B$).
  • ✅ Right: $B$ is the "strength" (Tesla), $\Phi$ is the "total amount" (Webers). Think of $B$ as the density of rain and $\Phi$ as the total water caught in a bucket.
• ❌ Wrong: Stating that a steady magnetic field induces an e.m.f.
  • ✅ Right: Only a changing magnetic flux linkage induces an e.m.f. If $\frac{d\Phi}{dt} = 0$, then $E = 0$.
• ❌ Wrong: Neglecting the number of turns $N$ when calculating the total e.m.f. for a coil.
  • ✅ Right: Faraday's law applies to the total flux linkage. Always multiply the flux through one loop by the total number of turns.

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Exam Tips

1. The "State and Explain" Lenz's Law Question: This is a classic 3 or 4-mark question. Always follow this logical path:
   • Identify the change: "The magnetic flux through the coil is increasing as the North pole approaches."
   • Apply Lenz's Law: "The induced current will flow in a direction to oppose this increase."
   • Describe the opposition: "A North pole is created at the end of the coil facing the magnet to repel it."
   • Conclude with Energy: "Work must be done to move the magnet against this force, which is converted into electrical energy."
2. Graphical Relationships:
   • If a graph of $N\Phi$ vs $t$ is a straight line (constant gradient), the $E$ vs $t$ graph is a horizontal line (constant value).
   • If $N\Phi$ is a sine curve, $E$ will be a cosine curve (because $E$ is the negative gradient).
   • The area under an $E$ vs $t$ graph represents the total change in magnetic flux linkage ($\Delta N\Phi$).
3. Search Coils: In problems involving search coils and oscilloscopes, the oscilloscope measures the induced e.m.f. If the oscilloscope shows a sine wave, it means the magnetic field in the primary coil is changing sinusoidally (AC).
4. Units Check: Always ensure $B$ is in Tesla (T), $A$ is in $\text{m}^2$, and $t$ is in seconds (s). If a value is given in $\text{mT}$ or $\mu\text{Wb}$, convert to standard form ($10^{-3}$ or $10^{-6}$) immediately to avoid power-of-ten errors.
5. Direction of Current: Use the Right-Hand Grip Rule to relate the direction of the induced current in a solenoid to the magnetic poles it creates. Your fingers curl in the direction of the current, and your thumb points toward the North pole.