Electromagnetism Revision Notes for Grade 11 NSC Matric Physical Sciences

Faraday's Law of Electromagnetic Induction

Introduction to electromagnetic induction

Electromagnetic induction is one of the most important discoveries in physics. Michael Faraday discovered that when you move a magnet near a wire, a voltage is generated across the wire. This discovery became the foundation for generating electricity practically.

The key insight was that the magnet must be moving to generate voltage. When the magnet stays still, no voltage appears. The voltage that appears during this process is called the induced emf (ε).

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Historical Context Faraday's discovery was revolutionary because it showed that magnetism and electricity are interconnected phenomena. This laid the groundwork for our modern understanding of electromagnetism and made possible the development of electric generators and motors.

When you set up a circuit loop with a sensitive ammeter and move a magnet up and down near it, the ammeter will register a current flowing through the circuit.

Understanding magnetic flux

Before we can fully understand Faraday's law, we need to understand magnetic flux. Think of magnetic flux as the amount of magnetic field that passes through a surface.

For a loop of area $A$ in the presence of a uniform magnetic field $B$ , the magnetic flux (φ) is defined as:

$\phi = BA \cos \theta$

Where:

$\theta$ = the angle between the magnetic field $B$ and the normal to the loop of area $A$
$A$ = the area of the loop
$B$ = the magnetic field strength

The SI unit of magnetic flux is the weber (Wb).

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Understanding the Angle θ The angle θ is crucial because it determines how much of the magnetic field actually passes through the surface. When the magnetic field is parallel to a surface, no field lines pass through it, so there's no contribution to the flux. Only the component of the field that is perpendicular to the surface contributes to the magnetic flux.

When the magnetic field is at an angle other than perpendicular, you can break it into components. The component perpendicular to the surface has magnitude $B \cos(\theta)$ , where $\theta$ is the angle between the normal and the magnetic field.

Faraday's law definition and formula

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DEFINITION: Faraday's Law of electromagnetic induction

The emf, ε, produced around a loop of conductor is proportional to the rate of change of the magnetic flux, φ, through the area, A, of the loop.

This can be stated mathematically as:

$\varepsilon = -N\frac{\Delta\phi}{\Delta t}$

Where:

$\phi = B \cdot A$ and $B$ is the strength of the magnetic field
$N$ is the number of circuit loops
$A$ is the magnetic field measured in units of teslas (T)
The minus sign indicates direction and that the induced emf tends to oppose the change in the magnetic flux

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The minus sign can be ignored when calculating magnitudes, but it's important for understanding the physical meaning of the law.

Faraday's Law relates induced emf to the rate of change of flux, which is the product of the magnetic field and the cross-sectional area through which the field lines pass.

Important considerations

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Critical Concept: Area Definition We use the area that the wire encloses, not the area of the wire itself. This means that if you bend wire into a circle, the area you use in flux calculations is the surface area of the circle, not the wire.

When the magnet is in the same plane as the circuit loop, there would be no current even if the magnet moved closer or further away. This happens because the magnetic field lines don't pass through the enclosed area but are parallel to it. The magnetic field lines must pass through the area enclosed by the circuit loop for an emf to be induced.

Direction of induced current

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Fundamental Principle The induced current opposes whatever change is taking place.

When you have a circuit loop with the south pole of a magnet moving closer, the magnitude of the field from the magnet increases. The induced emf will try to resist the field becoming stronger. Since the field is a vector, the current will flow in a direction so that the fields due to the current tend to cancel those from the magnet, keeping the resultant field constant.

To resist the change from an approaching south pole from above, the current must result in field lines that move away from the approaching pole. The induced magnetic field must therefore have field lines that go down on the inside of the loop. Using the Right Hand Rule: put your right thumb in the direction of one of the arrows on the circuit loop and notice what the field curls downwards into the area enclosed by the loop.

When the south pole moves away, the field from the magnet becomes weaker. The response from the induced current will be to set up a magnetic field that adds to the existing one from the magnet to resist it decreasing in strength.

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Memory Aid for Direction You can think of this as poles repelling each other. To resist an approaching south pole, the current creates a field that looks like another south pole on the side of the approaching south pole. Like poles repel, so you can think of the current setting up a south pole to repel the approaching south pole.

You can also use the Right Hand Rule by putting your fingers in the direction of the current to get your thumb to point in the direction of the field lines (or the north pole).

Direction of induced current in a solenoid

The approach for determining current direction in a solenoid is the same as described above. The only difference is that in a solenoid there are multiple loops of wire, so the magnitude of the induced emf will be different. The flux calculation uses the surface area of the solenoid multiplied by the number of loops.

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Solenoid vs Single Loop Remember: The directions of currents and associated magnetic fields can all be found using only the Right Hand Rule. When the fingers of the right hand are pointed in the direction of the magnetic field, the thumb points in the direction of the current. When the thumb is pointed in the direction of the magnetic field, the fingers point in the direction of the current.

The direction of the current will be such as to oppose the change. Here's how to determine the direction in different scenarios:

When a north pole approaches the solenoid: The current flows so that a north pole is established at the end of the solenoid closest to the approaching magnet to repel it.
When a north pole moves away from the solenoid: The current flows so that a south pole is established at the end of the solenoid closest to the receding magnet to attract it.
When a south pole moves away from the solenoid: The current flows so that a north pole is established at the end of the solenoid closest to the receding magnet to attract it.
When a south pole approaches the solenoid: The current flows so that a south pole is established at the end of the solenoid closest to the approaching magnet to repel it.

Induction applications

Electromagnetic induction has many practical applications. It's used in electrical generators which use mechanical power to move a magnetic field past coils of wire to generate voltage.

Self-induction occurs when a changing magnetic field is produced by changes in current through a wire, inducing a voltage along the length of that same wire. If the magnetic flux is enhanced by bending the wire into a coil and/or wrapping that coil around a material of high permeability, this effect becomes more intense. A device constructed to take advantage of this effect is called an inductor.

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Lenz's Law The induced current creates a magnetic field that opposes the change in the magnetic flux. This fundamental principle is known as Lenz's law and explains why there's a negative sign in Faraday's equation.

Worked example 1: Faraday's law

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Worked Example: Square Coil in Changing Magnetic Field

QUESTION Consider a flat square coil with 5 turns. The coil is 0.50 m on each side and has a magnetic field of 0.5 T passing through it. The plane of the coil is perpendicular to the magnetic field: the field points out of the page. Use Faraday's Law to calculate the induced emf, if the magnetic field increases uniformly from 0.5 T to 1 T in 10 s. Determine the direction of the induced current.

SOLUTION

Step 1: Identify what is required We are required to use Faraday's Law to calculate the induced emf.

Step 2: Write Faraday's Law $\varepsilon = -N\frac{\Delta\phi}{\Delta t}$

We know that the magnetic field is at right angles to the surface and so aligned with the normal. This means we do not need to worry about the angle that the field makes with the normal and $\phi = BA$ . The starting or initial magnetic field, $B_i$ , is given as is the final field magnitude, $B_f$ . We want to determine the magnitude of the emf so we can ignore the minus sign.

The area, $A$ , is the area of square coil.

Step 3: Solve Problem

$\varepsilon = N\frac{\Delta\phi}{\Delta t} = N\frac{\phi_f - \phi_i}{\Delta t} = N\frac{B_f A - B_i A}{\Delta t} = \frac{N \cdot A(B_f - B_i)}{\Delta t}$
$\varepsilon = \frac{(5)((0.50)^2(1 - 0.50))}{10} = \frac{(5)(0.25)(0.5)}{10} = 0.0625 \text{ V}$

The induced current is anti-clockwise as viewed from the direction of the increasing magnetic field.

Worked example 2: Faraday's law

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Worked Example: Finding Solenoid Radius

QUESTION Consider a solenoid of 9 turns with unknown radius, $r$ . The solenoid is subjected to a magnetic field of 0.12 T. The axis of the solenoid is parallel to the magnetic field. When the field is uniformly switched to 12 T over a period of 2 minutes an emf with a magnitude of -0.3 V is induced. Determine the radius of the solenoid.

SOLUTION

Step 1: Identify what is required We are required to determine the radius of the solenoid. We know that the relationship between the induced emf and the field is governed by Faraday's law which includes the geometry of the solenoid. We can use this relationship to find the radius.

Step 2: Write Faraday's Law $\varepsilon = -N\frac{\Delta\phi}{\Delta t}$

We know that the magnetic field is at right angles to the surface and so aligned with the normal. This means we do not need to worry about the angle the field makes with the normal and $\phi = BA$ . The starting or initial magnetic field, $B_i$ , is given as is the final field magnitude, $B_f$ . We can drop the minus sign because we are working with the magnitude of the emf only.

The area, $A$ , is the surface area of the solenoid which is $\pi r^2$ .

Step 3: Solve Problem

$\varepsilon = N\frac{\Delta\phi}{\Delta t} = N\frac{\phi_f - \phi_i}{\Delta t} = N\frac{B_f A - B_i A}{\Delta t} = \frac{N \cdot A(B_f - B_i)}{\Delta t}$
$\varepsilon = \frac{N((\pi r^2)(12 - 0.12))}{120}$
$(0.30) = \frac{(9)((\pi r^2)(12 - 0.12))}{120}$
$r^2 = \frac{(0.30)(120)}{(9)\pi(12 - 0.12)} = \frac{36}{(9)\pi(11.88)} = 0.107175$
$r = 0.32 \text{ m}$

The solenoid has a radius of 0.32 m.

Worked example 3: Faraday's law

lightbulbExample

Worked Example: Circular Coil at an Angle

QUESTION Consider a circular coil of 4 turns with radius $3 \times 10^{-2}$ m. The solenoid is subjected to a varying magnetic field that changes uniformly from 0.4 T to 3.4 T in an interval of 27 s. The axis of the solenoid makes an angle of 35° to the magnetic field. Find the induced emf.

SOLUTION

Step 1: Identify what is required We are required to use Faraday's Law to calculate the induced emf.

Step 2: Write Faraday's Law $\varepsilon = -N\frac{\Delta\phi}{\Delta t}$

We know that the magnetic field is at an angle to the surface normal. This means we must account for the angle that the field makes with the normal and $\phi = BA \cos(\theta)$ . The starting or initial magnetic field, $B_i$ , is given as is the final field magnitude, $B_f$ . We want to determine the magnitude of the emf so we can ignore the minus sign.

The area, $A$ , will be $\pi r^2$ .

Step 3: Solve Problem

$\varepsilon = N\frac{\Delta\phi}{\Delta t} = N\frac{\phi_f - \phi_i}{\Delta t} = N\frac{B_f A \cos(\theta) - B_i A \cos(\theta)}{\Delta t}$
$\varepsilon = \frac{N(A \cos(\theta)(B_f - B_i))}{\Delta t}$
$\varepsilon = \frac{(4)((\pi(0.03)^2 \cos(35°))(3.4 - 0.4))}{27} = 1.03 \times 10^{-3} \text{ V}$

The induced current is anti-clockwise as viewed from the direction of the increasing magnetic field.

Real-life applications

Faraday's Law is used in the operation of many everyday devices:

Induction stoves - use changing magnetic fields to heat cookware
Tape players - read magnetic information using electromagnetic induction
Metal detectors - detect changes in magnetic fields caused by metal objects
Transformers - transfer electrical energy between circuits using changing magnetic fields

These applications demonstrate how Faraday's discovery continues to impact our daily lives through the practical generation and manipulation of electrical energy.

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Key Points to Remember:

Electromagnetic induction occurs when a changing magnetic field generates an induced emf in a conductor
Faraday's Law states that induced emf equals the negative rate of change of magnetic flux: $\varepsilon = -N\frac{\Delta\phi}{\Delta t}$
Magnetic flux depends on field strength, area, and the angle between field and surface: $\phi = BA \cos \theta$
The direction of induced current always opposes the change causing it (Lenz's law)
Use the Right Hand Rule to determine current direction in electromagnetic induction problems
The SI unit of magnetic flux is the weber (Wb)

Electromagnetism (Grade 11 NSC Matric Physical Sciences): Revision Notes

Faraday's Law of Electromagnetic Induction

Introduction to electromagnetic induction

Understanding magnetic flux

Faraday's law definition and formula

Important considerations

Direction of induced current

Direction of induced current in a solenoid

Induction applications

Worked example 1: Faraday's law

Worked example 2: Faraday's law

Worked example 3: Faraday's law

Real-life applications

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