Electromagnetic Induction (Section 3.10.4)
Hello! Welcome to one of the most fundamental chapters in modern electrical science: Electromagnetic Induction. This is where we learn how to turn movement and magnetism into usable electricity—the principle behind power generation, credit card readers, and wireless charging.
In this section, we move from magnetic fields affecting currents (motors) to changing magnetic fields creating currents (generators). Get ready to meet the foundational laws established by Faraday and Lenz!
Review of Magnetic Flux and Flux Linkage
We need to remember the key quantity that must change for induction to happen: Magnetic Flux Linkage (\(N\Phi\)).
- Magnetic Flux (\(\Phi\)): A measure of the total magnetic field passing through a surface (measured in Webers, Wb).
- Flux Linkage (\(N\Phi\)): If a coil has \(N\) turns, the total flux linking the coil is \(N\) times the flux through one turn.
1. Simple Experimental Phenomena
Experimental observation shows that an e.m.f. (electromotive force, or voltage) is induced in a circuit if, and only if, the magnetic flux linkage through the circuit is changing.
Here are the three ways to create a changing flux linkage, \(\Delta (N\Phi)\):
- Changing Field Strength (\(B\)): Moving a magnet toward or away from a coil. When the magnet stops, the current stops.
- Changing Area (\(A\)): Expanding or contracting a loop within a steady magnetic field.
- Changing Angle (\(\theta\)): Rotating a coil within a steady magnetic field (the basis of the generator).
Analogy: Think of pumping a swing. An e.m.f. is only induced when you are actively moving the components. If you hold the magnet steady, even if the field is strong, no electricity is produced.
Key Takeaway
No change in flux linkage means no induced e.m.f. Static fields don't generate electricity.
2. Faraday's Law: The Magnitude of the e.m.f.
Faraday's Law quantifies the size of the induced e.m.f. (\(\mathcal{E}\)).
The Definition
Faraday's Law of Induction states that the magnitude of the induced e.m.f. is directly proportional to the rate of change of magnetic flux linkage.
The equation, where \(\Phi\) represents the total flux linkage (\(N\Phi\)):
$$ \mathcal{E} = \frac{\Delta \Phi}{\Delta t} $$- \(\mathcal{E}\): Induced e.m.f. (V)
- \(\Delta \Phi\): Change in flux linkage (Wb)
- \(\Delta t\): Time taken for the change (s)
What This Means for Practice
If you want to induce a larger e.m.f., you must:
- Use a coil with a greater number of turns (\(N\)) or a larger area (\(A\)).
- Increase the magnetic field strength (\(B\)).
- Change the flux linkage faster (decrease \(\Delta t\)).
Working with Graphs: If you are given a graph of flux linkage versus time, the induced e.m.f. is simply the gradient of the graph at any point. A steeper graph (a rapid change) means a larger induced e.m.f.
Key Takeaway
The voltage induced depends on the speed of the change. Push the magnet harder, get more voltage.
3. Lenz's Law: The Direction of the e.m.f.
While Faraday tells us "how much," Lenz's Law tells us "which way" the current flows. It provides the necessary direction for the induced e.m.f.
The Principle of Opposition
Lenz's Law states that the induced e.m.f. (or induced current) acts in such a direction as to oppose the change in magnetic flux linkage that produced it.
Encouragement: Don't worry if this sounds counter-intuitive! This opposition is essential because it guarantees the conservation of energy. You have to do work against the opposing force to induce the current. Energy is never created for free.
Understanding the Opposition
- If you push a North pole towards a coil (flux is increasing), the coil generates an induced North pole to repel the approaching magnet.
- If you pull the North pole away from the coil (flux is decreasing), the coil generates an induced South pole to attract the magnet and try to prevent it from leaving.
In the combined equation, the negative sign mathematically represents Lenz's Law:
$$ \mathcal{E} = - \frac{\Delta \Phi}{\Delta t} $$
We usually ignore the negative sign when calculating the magnitude, but its presence signifies that the e.m.f. is always in the direction that opposes the flux change.
Key Takeaway
Lenz's Law is all about opposition. The induced current always tries to keep things exactly as they were, fighting the external change.
4. Induction in Specific Systems and Applications
4.1 Straight Conductor Moving in a Field
Consider a straight conducting rod of length \(L\) moving with velocity \(v\) perpendicular to a uniform magnetic field \(B\).
The induced e.m.f. is given by:
$$ \mathcal{E} = BLv $$
This is an important application, as it forms the basis of simple linear generators and explains the concept of motional e.m.f.
To determine the direction of the induced current or e.m.f., we use Fleming's Right Hand Rule (Generator Rule):
- Thumb: Motion (Direction of conductor movement)
- First Finger: Field (Direction of magnetic field B)
- Second Finger: Current (Direction of induced current I)
4.2 Emf Induced in a Rotating Coil (AC Generators)
When a coil rotates at a constant angular velocity \(\omega\) in a uniform magnetic field \(B\), the induced e.m.f. changes constantly, producing an AC voltage.
The e.m.f. induced in a coil with \(N\) turns and area \(A\) rotating uniformly is:
$$ \mathcal{E} = BAN\omega \sin \omega t $$The voltage output follows a sine wave pattern because:
- When the coil is cutting the flux lines perpendicularly (highest rate of change), \(\sin \omega t = 1\), and \(\mathcal{E}\) is maximum: \(\mathcal{E}_{max} = BAN\omega\).
- When the coil is moving parallel to the flux lines (zero rate of change), \(\sin \omega t = 0\), and \(\mathcal{E}\) is zero.
4.3 Production of Eddy Currents
When a bulk conductor (like a solid metal plate) experiences a changing magnetic flux, loops of current are induced within the material itself. These circulating currents are known as Eddy Currents.
Consequences (Lenz's Law): These currents generate a magnetic field that opposes the motion causing them, creating a strong damping or braking force. This is used deliberately in electromagnetic braking systems.
Avoiding Energy Loss: In devices like transformers, motors, and generators, we want to maximise the useful e.m.f., not generate wasteful heat from eddy currents. To limit this power loss (\(P=I^2R\)), the iron cores are made of thin, insulated sheets called laminations.
Laminating the core drastically increases the resistance \(R\) that the circulating eddy currents must overcome, thereby reducing their magnitude \(I\) and minimising heat loss.
Key Takeaway
Induction can be achieved linearly (\(\mathcal{E}=BLv\)) or rotationally (\(\mathcal{E}=BAN\omega \sin\omega t\)). Eddy currents are internal currents induced in bulk materials, useful for braking but wasteful if not controlled by lamination.
Did you know?
The induction cooker in your kitchen uses high-frequency AC current to create a rapidly changing magnetic field. This field induces huge eddy currents directly into the base of the magnetic metal pan, heating the pan itself, while leaving the cooker surface relatively cool. It's a highly efficient application of Faraday’s Law!