Faraday’S Law Of Induction- Induced Emf - Motional Electromotive Force Example 2

Faraday’s Law of Induction

Explains the relationship between a changing magnetic field and the induced electromotive force (emf)

Induced Electromotive Force (emf)

Induced emf is produced when there is a change in the magnetic field through a closed loop of wire
This change in magnetic field can be caused by motion of the wire or a changing magnetic field nearby

Motional Electromotive Force

Motional emf is generated when a conductor moves across a magnetic field
The magnitude of motional emf is given by the equation:

$\varepsilon = B \cdot L \cdot v \cdot \sin \theta$
Where:
- $\varepsilon$ is the induced emf
- B is the magnetic field strength
- L is the length of the conductor
- v is the velocity of the conductor
- $\theta$ is the angle between the velocity and magnetic field vectors

Faraday’s Law of Induction (General Form)

The induced emf in a closed loop of wire is directly proportional to the rate of change of magnetic flux through the loop
Mathematically, this can be expressed as:

$\varepsilon = - \frac{d\Big(BA\Big)}{dt}$
Where:
- $\varepsilon$ is the induced emf
- $B$ is the magnetic field strength
- $A$ is the area of the loop
- $\frac{d}{dt}$ denotes the derivative with respect to time

Faraday’s Law of Induction (Simplified Form)

For a loop with a constant area and where the magnetic field is perpendicular to the loop, the induced emf can be expressed as:

$\varepsilon = - A \frac{d\Big(B\Big)}{dt}$

Example 1

A circular loop with a radius of 0.1 m is placed in a magnetic field that is changing at a rate of 0.5 T/s
Calculate the induced emf in the loop
Solution:
- Given:
  - $\frac{d\Big(B\Big)}{dt}$ = 0.5 T/s
  - $A$ = $\pi \cdot r^2$ = $\pi \cdot 0.1^2$ = 0.0314 m^2
- Using the simplified form of Faraday’s Law:
  - $\varepsilon = - A \frac{d\Big(B\Big)}{dt}$
  - $\varepsilon = - 0.0314 \cdot 0.5$
  - $\varepsilon = -0.0157 V$

Lenz’s Law

Lenz’s Law provides the direction of the induced current in a closed loop when there is a changing magnetic field
It states that the direction of the induced current is such that it opposes the change in magnetic flux that caused it

Lenz’s Law and Motional emf

Lenz’s Law can also be applied to motional emf situations
The direction of the induced current is such that it creates a magnetic field that opposes the motion of the conductor

Example 2

A rod of length 0.2 m is moving with a velocity of 10 m/s perpendicular to a magnetic field of 0.5 T
Calculate the induced emf and the direction of the induced current, assuming the magnetic field is perpendicular to the rod
Solution:
- Given:
  - $\varepsilon = B \cdot L \cdot v \cdot \sin \theta$
  - $\theta$ = 90 degrees (perpendicular)
- Substituting the given values:
  - $\varepsilon = 0.5 \cdot 0.2 \cdot 10 \cdot \sin 90 = 1 V$

Summary

Faraday’s Law of Induction explains how a change in magnetic field induces an emf in a closed loop of wire
Motional emf is generated when a conductor moves across a magnetic field
Lenz’s Law provides the direction of the induced current and states that it opposes the change in magnetic flux.

Faraday’s Law Of Induction

Faraday’s Law explains the relationship between a changing magnetic field and the induced electromotive force (emf).
It states that the induced emf in a closed loop of wire is directly proportional to the rate of change of magnetic flux through the loop.
This law is one of the fundamental principles of electromagnetism.

Induced Electromotive Force (emf)

Induced emf is produced when there is a change in the magnetic field through a closed loop of wire.
It can be caused by either a motion of the wire or a changing magnetic field nearby.
This induced emf can lead to the generation of electric currents in the wire.

Motional Electromotive Force

Motional emf is a type of induced emf that is generated when a conductor moves across a magnetic field.
The magnitude of motional emf is given by the equation: ε = B * L * v * sin(θ)
ε represents the induced emf, B is the magnetic field strength, L is the length of the conductor, v is the velocity of the conductor, and θ is the angle between the velocity and magnetic field vectors.

Faraday’s Law Of Induction (General Form)

The general form of Faraday’s Law of Induction states that the induced emf in a closed loop is equal to the negative rate of change of magnetic flux through the loop.
Mathematically, it can be expressed as: ε = -(d/dt)(BA)
Here, ε represents the induced emf, B is the magnetic field strength, and A is the area of the loop.

Faraday’s Law Of Induction (Simplified Form)

In certain cases, where the magnetic field is perpendicular to the loop and the area is constant, Faraday’s Law can be simplified.
The simplified form states that the induced emf is equal to the negative product of the loop area and the rate of change of magnetic field.
Mathematically, it is given by: ε = -A(dB/dt)

Lenz’s Law

Lenz’s Law provides the direction of the induced current in a closed loop when there is a changing magnetic field.
According to Lenz’s Law, the direction of the induced current is such that it opposes the change in magnetic flux that caused it.
This law helps to determine the polarity of induced currents and their effect on magnetic fields.

Lenz’s Law and Motional emf

Lenz’s Law can also be applied to motional emf situations.
The direction of the induced current due to motional emf is such that it creates a magnetic field that opposes the motion of the conductor.
This principle is consistent with the conservation of energy, as the induced current works against the force causing the motion.

Example: Motional emf

Consider a wire of length 0.3 m moving at a velocity of 5 m/s across a magnetic field of magnitude 0.2 T.
The angle between the velocity and magnetic field vectors is 60 degrees.
Using the formula for motional emf, we can calculate the induced emf: ε = B * L * v * sin(θ)
Substituting the given values, we get: ε = 0.2 * 0.3 * 5 * sin(60) = 0.15 V.

Example: Lenz’s Law

Suppose there is a loop of wire in which the magnetic field increases.
According to Lenz’s Law, the induced current in the loop will oppose this increase in magnetic field.
To achieve this, the induced current creates a magnetic field that opposes the change in the external magnetic field.
This principle can be observed in various electromagnetic applications.

Summary

Faraday’s Law of Induction explains the relationship between changing magnetic fields and induced emf.
Motional emf is a type of induced emf generated when a conductor moves across a magnetic field.
Faraday’s Law and Lenz’s Law are fundamental principles in electromagnetism that help us understand the direction and magnitude of induced currents in closed loops.

Induced emf (Electromagnetic Induction)

Induced emf is the electromotive force produced by a changing magnetic field through a closed loop of wire or a coil
It can be caused by various factors including:
- Moving a wire across a magnetic field
- Changing the magnetic field strength through a loop
- Rotating a coil in a magnetic field

Induced emf (Equation)

The equation for induced emf can be derived from Faraday’s Law of Induction:
- ε = - N * d(Φ) / dt
Where:
- ε represents the induced emf
- N is the number of turns in the coil
- d(Φ) / dt is the rate of change of magnetic flux through the loop or coil

Motional Electromotive Force (Example)

Consider a wire of length 0.5 m moving at a velocity of 8 m/s perpendicular to a magnetic field of 0.4 T
Calculate the induced emf in the wire
Solution:
- Given:
  - L = 0.5 m
  - v = 8 m/s
  - B = 0.4 T
  - θ = 90 degrees (perpendicular)
- Using the equation ε = B * L * v * sin(θ):
  - ε = 0.4 * 0.5 * 8 * sin(90) = 1.6 V

Changing Magnetic Field Strength (Example)

Suppose the magnetic field through a coil of 100 turns changes from 0.2 T to 0.6 T in 0.5 seconds
Calculate the induced emf in the coil
Solution:
- Given:
  - N = 100 turns
  - B1 = 0.2 T
  - B2 = 0.6 T
  - Δt = 0.5 s
- Using the equation ε = -N * d(Φ) / dt:
  - Δ(Φ) = B2 * A - B1 * A
  - ε = -N * Δ(Φ) / Δt
  - ε = -100 * (0.6 - 0.2) * A / 0.5
  - ε = -40 V

Rotating Coil in Magnetic Field

When a coil is rotated in a magnetic field, the magnetic flux through the coil changes, inducing an emf
The emf is maximum when the coil is parallel or perpendicular to the magnetic field lines
The emf is zero when the coil is at an angle of 45 or 135 degrees to the magnetic field lines

Examples of Practical Applications

Generators: Induced emf is used to convert mechanical energy into electrical energy in power generators
Transformers: Induced emf is utilized in transformers to transfer electrical energy between different voltage levels
Electric Motors: Induced emf is responsible for the motion in electric motors
Induction Cooktops: Induced emf is utilized to generate heat for cooking

Lenz’s Law (Recap)

Lenz’s Law states that the direction of the induced current in a closed loop is such that it opposes the change in magnetic flux that caused it
This law is consistent with the conservation of energy principle

Lenz’s Law Application: Transformer

In a transformer, Lenz’s Law ensures that the secondary emf opposes the change in the magnetic field from the primary coil
This opposition allows for efficient energy transfer between the coils

Lenz’s Law Application: Electric Motor

In an electric motor, Lenz’s Law causes a torque to be exerted on the rotor, allowing the motor to rotate
The induced current in the rotor opposes the change in the magnetic field, creating a driving force

Summary

Induced emf is the electromotive force produced by a changing magnetic field through a closed loop of wire or a coil
Motional emf is induced when a wire moves across a magnetic field, while changing magnetic field strength and rotating coils also lead to induced emf
Lenz’s Law determines the direction of the induced current, opposing the change in magnetic flux
These principles have practical applications in generators, transformers, electric motors, and induction cooktops