Faraday's Law

Learning Objectives

• At the end of the lesson, 80% of the students should be able to:

  1. Explain Faraday’s Law based on its importance in electromagnetism.

  2. Calculate the induced emf in a closed loop due to a time-varying magnetic flux using Faraday’s Law.

  3. Value the importance of Faraday’s Law in our daily lives.

Michael Faraday

• An English physicist and chemist.

• Contributed significantly to the understanding of electromagnetism through numerous experiments.

• First to generate an electric current from a magnetic field.

Faraday's Law and Electromagnetic Induction

• Electromagnetic Induction: The process of using magnetic fields to produce voltage; in a closed circuit, this results in current.

Principle of Faraday’s Law

• Faraday's Law of Induction: A fundamental law in electromagnetism predicting how a magnetic field interacts with an electric circuit to produce an electromotive force (emf), known as electromagnetic induction.

  • This law is foundational for the operation of transformers, inductors, electrical motors, generators, and solenoids.

Magnetic Flux

• Defined as the number of magnetic field lines passing through a closed surface: $ext{Flux} ( ext{Φ}) = B imes A imes ext{cos}( heta)$

  • Where:

  • $B$ = magnetic field strength (T)

  • $A$ = area (m²)

  • $heta$ = angle with the perpendicular to the area.

• Units: Weber (Wb), equivalent to $ext{T} imes ext{m}^2$.

Understanding Magnetic Flux

• The magnetic flux can be represented as:

  • $ext{Φ} = B A ext{cos}( heta)$

  • As indicated, $B$ represents the component of the magnetic field perpendicular to the area.

Calculating the Perpendicular Component of the Magnetic Field

• Perpendicular component can be represented as: $B = B ext{cos}( heta)$

  • If $heta = 0$: field is fully perpendicular, hence $B = 1$

  • If $heta = 90$: field is parallel, hence $B_{\bot} = 0$

  • If $heta = 60$: half of the field passes through, hence $B = 0.5$

Electromotive Force (emf)

• Defined as the voltage created when a changing magnetic field induces current in a circuit.

• Units: Volts (V).

• Equation for induced emf due to magnetic flux changes:
  $ext{emf} = - rac{d ext{Φ}}{dt}$

Factors Affecting Induced emf

• Faraday’s experiments revealed that emf induced by changes in magnetic flux depends on several factors:

  1. The change in flux ($riangle ext{Φ}$).

  2. The change in time ($riangle t$); the smaller this change, the greater the emf induced (inversely proportional).

  3. The number of turns in the coil ($N$); an emf will be produced that is $N$ times greater than that for a single coil (directly proportional).

Determining Induced Voltage

• Induced voltage (emf) can be influenced by:

  1. Increasing the number of turns of wire in the coil.

  2. Increasing the speed of relative motion between the coil and the magnetic field.

  3. Increasing the strength of the magnetic field.

Sample Problem 1

• Problem: A circular wire loop of radius $r = 0.10 ext{ m}$ is placed in a uniform magnetic field of strength $B = 0.50 ext{ T}$ at an angle of $heta = 60 ext{°}$ to the normal of the loop's plane. Find the magnetic flux through the loop.

  • Given:

  • Radius of loop: $r = 0.10 ext{ m}$

  • Magnetic field strength: $B = 0.50 ext{ T}$

  • Angle: $heta = 60 ext{°}$

  • Required: Magnetic flux ($ext{Φ}$)

Calculation Steps

1. Calculate the area of the loop:
   $A = ext{π}r^{2} = 3.1416 imes (0.10 ext{ m})^{2} = 0.0314 ext{ m}^{2}$

2. Find the perpendicular component of the magnetic field:
   $B_{\bot} = B ext{cos}( heta) = 0.50 ext{ T} imes ext{cos}(60) = 0.25 ext{ T}$

3. Calculate magnetic flux:
   $ext{Φ} = B_{\bot} A = 0.25 ext{ T} imes 0.0314 ext{ m}^{2} = 0.00785 ext{ Wb}$

• Final Answer: The magnetic flux through the loop is $7.85 imes 10^{-3} ext{ Wb}$.

Sample Problem 2

• Problem: Calculate the magnitude of the induced emf when a magnet is thrust into a coil with a radius of $6.00 ext{ cm}$. The magnetic field changes from $0.0500 ext{ T}$ to $0.250 ext{ T}$ over $0.100 ext{ s}$.

  • Given:

  • $r = 0.0600 ext{ m}$

  • Initial Magnetic Field: $B_i = 0.0500 ext{ T}$

  • Final Magnetic Field: $B_f = 0.250 ext{ T}$

  • Change in time: $riangle t = 0.100 ext{ s}$

  • Number of turns: $N = 1$

  • Required: Magnitude of induced emf ($E$).

Calculation Steps

1. Calculate the area of the coil:
   $A = ext{π}r^{2} = 3.1416 imes (0.0600 ext{ m})^{2} = 0.0113 ext{ m}^{2}$

2. Calculate change in magnetic field component:
   $riangle B = B<em>f - B</em>i = 0.250 ext{ T} - 0.0500 ext{ T} = 0.200 ext{ T}$

3. Calculate change in magnetic flux:
   $riangle ext{Φ}_B = A riangle B = (0.0113 ext{ m}^{2})(0.200 ext{ T}) = 0.00226 ext{ Wb}$

4. Calculate induced emf:
   $E = - rac{ riangle ext{Φ}}{ riangle t} = rac{0.00226 ext{ Wb}}{0.100 ext{ s}} = 0.0226 ext{ V} = 22.6 ext{ mV}$

• Final Answer: Magnitude of the induced emf is $E = 2.26 imes 10^{-2} ext{ mV}$.

Exercise Problems

• Problem 1: A rectangular loop with dimensions $0.20 ext{ m} imes 0.10 ext{ m}$ in a uniform magnetic field of $B = 0.40 ext{ T}$, oriented perpendicular to the plane of the loop. Calculate the magnetic flux.

  • Given:

  • Length: $l = 0.20 ext{ m}$

  • Width: $w = 0.10 ext{ m}$

  • Magnetic field: $B = 0.40 ext{ T}$

  • Angle: $heta = 0$

Solution Steps for Problem 1

1. Calculate the area of the loop:
   $A = l imes w = (0.20 ext{ m})(0.10 ext{ m}) = 0.020 ext{ m}^{2}$

2. Determine the angle factor:

   • $ext{cos}(0) = 1$

3. Calculate magnetic flux:
   $ext{Φ}_B = B A ext{cos}( heta) = (0.40 ext{ T})(0.020 ext{ m}^{2})(1) = 0.0080 ext{ Wb}$

• Final Answer: The magnetic flux through the loop is $8.0 imes 10^{-3} ext{ Wb}$.

• Problem 2: A single-turn circular loop of radius $r = 0.05 ext{ m}$ is in a magnetic field that increases from $0.10 ext{ T}$ to $0.30 ext{ T}$ over $0.20 ext{ s}$. Calculate the induced emf.

  • Given:

  • $r = 0.05 ext{ m}$

  • Initial magnetic field: $B_i = 0.10 ext{ T}$

  • Final magnetic field: $B_f = 0.30 ext{ T}$

  • Change in time: $riangle t = 0.20 ext{ s}$

  • Number of turns: $N = 1$

Solution Steps for Problem 2

1. Calculate the area of the loop:
   $A = ext{π} r^{2} = 3.1416 imes (0.05 ext{ m})^{2} = 0.00785 ext{ m}^{2}$

2. Calculate change in magnetic field:
   $riangle B = B<em>f - B</em>i = 0.30 ext{ T} - 0.10 ext{ T} = 0.20 ext{ T}$

3. Calculate change in magnetic flux:
   $riangle ext{Φ}_B = A riangle B = (0.00785 ext{ m}^{2})(0.20 ext{ T}) = 0.00157 ext{ Wb}$

4. Calculate induced emf:
   $E = - rac{ riangle ext{Φ}}{ riangle t} = rac{0.00157 ext{ Wb}}{0.20 ext{ s}} = 0.00785 ext{ V} = 7.85 ext{ mV}$

• Final Answer: The induced emf is $7.85 imes 10^{-3} ext{ mV}$.

Applications of Faraday's Law

• Faraday's Law enables various technological applications that have transformed human existence post-discovery, such as:

1. Electric Generators

• Operate on the principle that rotating a coil in a magnetic field induces current, facilitating large-scale electricity production and powering homes and industries.

2. Induction Stove

• Utilizes alternating current in a coil to create a changing magnetic field inducing currents that heat metal pans directly, providing fast, efficient cooking without open flames.

3. Transformers

• Work by inducing voltage in secondary coils through changing magnetic flux in primary coils, crucial for safe long-distance electricity transmission.

4. Communication Devices

• Antennas and microphones convert signals into electrical currents and vice versa, enhancing global communication and media dissemination.

5. Magnetic Resonance Imaging (MRI)

• Employs rapidly oscillating magnetic fields to induce signals in hydrogen atoms, converted into images for non-invasive medical assessments.

Reflection Questions

• 1. Identify an everyday device or machine that operates on Faraday’s Law.

• 2. Discuss how Faraday’s Law enhances convenience in daily life.

• 3. Explain the significance of recognizing the role of Faraday’s Law within technology and industry.