Prepare for University Studies & Career Advancement

Electrodynamics

Electrodynamics

Electrodynamics is a central pillar of modern physics, explaining how electric charges produce and respond to time-varying electric and magnetic fields. Unlike electrostatics or magnetostatics, which deal with stationary or steady-state conditions, electrodynamics focuses on dynamic situations where fields change with time. The discipline lays the groundwork for many technologies and natural phenomena, from radio transmission to electromagnetic radiation in space.

The study begins by understanding how currents and charges interact through electrical circuits, forming the basis for more advanced field analysis. Concepts such as electromagnetic induction reveal how time-varying magnetic fields can induce electric currents, an effect essential to transformers, electric motors, and power generation. These dynamic interactions also produce electromagnetic waves, which propagate energy through space and form the basis of all wireless communication.

Understanding the field-based nature of electrodynamics requires grasping the roles played by both magnetic fields and electric fields. When coupled with the behavior of charged particles in motion, these fields influence the structure of plasmas, studied in plasma physics, and the behavior of conductive fluids, explored in magnetohydrodynamics (MHD). Electrodynamics even contributes to understanding superconducting systems where resistance vanishes entirely, a focus of superconductivity.

At deeper theoretical levels, the unification of electrodynamics with quantum principles gives rise to quantum electrodynamics (QED), one of the most precisely tested theories in all of science. Electrodynamics also serves as a bridge into related domains such as light and optics, where the electromagnetic wave nature of light becomes evident.

Applications abound in fields like atmospheric and environmental optics and bio-optics, which rely on electrodynamic principles for interpreting phenomena such as scattering and absorption. Fiber-optic networks, discussed in fiber optics, depend on controlled electromagnetic propagation through glass strands. Techniques used in geometrical optics and laser optics similarly exploit electric and magnetic field behavior to manipulate light paths and coherence.

Modern breakthroughs in nonlinear optics and photonics continue to push the boundaries of what is possible in electrodynamic manipulation, including optical computing and ultra-fast communications. At the same time, the quantum aspects of these interactions, elaborated in quantum optics, deepen our understanding of light-matter interactions.

From lenses in visual optics to interference in wave optics, electrodynamics is embedded in our optical technologies. These ideas also permeate the broader field of modern physics, where field theory, relativity, and quantum models converge to form unified explanations of nature.

Thus, electrodynamics not only serves as a fundamental part of the electricity & magnetism domain but also as a gateway to a wide range of interdisciplinary applications and theoretical innovations in contemporary science and technology.

The image illustrates Electrodynamics, depicting dynamic electric and magnetic field lines, electromagnetic waves propagating through space, and energy transfer through conductive materials,
The image illustrates Electrodynamics, depicting dynamic electric and magnetic field lines, electromagnetic waves propagating through space, and energy transfer through conductive materials,

Table of Contents

Key Concepts in Electrodynamics

Electric and Magnetic Fields

  • Electric Field (𝐸⃗ ): A vector field that exerts a force on electric charges.
  • Magnetic Field (𝐵⃗ ): A vector field that exerts a force on moving charges and magnetic materials.

Lorentz Force

The total force on a charged particle moving in electric and magnetic fields is given by the Lorentz force equation:

𝐹⃗=𝑞(𝐸⃗+𝑣⃗×𝐵⃗)

Where:

  • 𝑞 = Electric charge
  • 𝑣⃗  = Velocity of the particle
  • 𝐸⃗  = Electric field
  • 𝐵⃗  = Magnetic field

Maxwell’s Equations

James Clerk Maxwell unified electricity and magnetism through four fundamental equations:

  1. Gauss’s Law for Electricity (Electric charges produce electric fields):

    ∇⋅𝐸⃗=𝜌𝜀0

    Gauss’s Law for Magnetism (No magnetic monopoles exist):

    ∇⋅𝐵⃗=0\nabla \cdot 𝐵⃗ = 0

    Faraday’s Law of Induction (A changing magnetic field induces an electric field):

    ∇×𝐸⃗=−∂𝐵⃗∂t\nabla × 𝐸⃗ = -\frac{\partial 𝐵⃗ }{\partial t}

  2. Ampère-Maxwell Law (Electric current and changing electric fields produce magnetic fields):

    ∇×𝐵⃗=μ0𝐽⃗+μ0ε0∂𝐸⃗∂t\nabla × 𝐵⃗ = \mu_0 𝐽⃗ + \mu_0 \varepsilon_0 \frac{\partial 𝐸⃗ }{\partial t}

Electromagnetic Waves

Maxwell’s equations predict that oscillating electric and magnetic fields can propagate through space as electromagnetic (EM) waves. The speed of these waves in a vacuum is given by:

c=1μ0ε0≈3×108 m/s

EM waves are transverse waves, meaning that the electric and magnetic fields are perpendicular to each other and to the direction of propagation.

Energy in Electrodynamics

Poynting Vector

The Poynting vector (𝐒⃗ ) represents the directional energy flux (power per unit area) of an electromagnetic field:

S⃗=𝐸⃗×𝐻⃗

Where:

𝐻⃗=𝐵⃗μ0

Energy Density

The energy stored in electric and magnetic fields is given by:

u=12ε0𝐸2+12𝐵2μ0u = \frac{1}{2} \varepsilon_0 𝐸^2 + \frac{1}{2} \frac{𝐵^2}{\mu_0}

Electric and Magnetic Fields

  • Electric Field : A vector field that exerts force on electric charges.
  • Magnetic Field : A vector field that exerts force on moving charges and magnetic materials.

Lorentz Force

The total force on a charged particle moving in electric and magnetic fields is given by the Lorentz force: F⃗=q(E⃗+v⃗×B⃗)\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})

Where:

  • 𝑞 is the electric charge.
  • 𝑣⃗ is the velocity of the particle.
  • 𝐸⃗ is the electric field.
  • 𝐵⃗ is the magnetic field.

Maxwell’s Equations

James Clerk Maxwell unified electricity and magnetism through four fundamental equations:
  1. Gauss’s Law for Electricity (Electric charges produce electric fields): ∇⋅E⃗=ρε0Gauss’s Law for Magnetism (No magnetic monopoles exist): ∇⋅B⃗=0\nabla \cdot \vec{B} = 0
  2. Faraday’s Law of Induction (A changing magnetic field induces an electric field): ∇×E⃗=−∂B⃗∂t\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}
  3. Ampère-Maxwell Law (Electric current and changing electric fields produce magnetic fields): ∇×B⃗=μ0J⃗+μ0ε0∂E⃗∂t

Electromagnetic Waves

Maxwell’s equations predict that oscillating electric and magnetic fields can propagate through space as electromagnetic (EM) waves. The speed of these waves in a vacuum is: c=1μ0ε0≈3×108 m/sc = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3 \times 10^8 \, \text{m/s} EM waves are transverse, with electric and magnetic fields perpendicular to each other and to the direction of propagation.

Energy in Electrodynamics

Poynting Vector

The Poynting vector represents the directional energy flux (power per unit area) of an electromagnetic field: S⃗=E⃗×H⃗\vec{S} = \vec{E} \times \vec{H} Where H⃗=B⃗μ0\vec{H} = \frac{\vec{B}}{\mu_0}

Energy Density

The energy stored in electric and magnetic fields is given by: u=12ε0E2+12B2μ0u = \frac{1}{2} \varepsilon_0 E^2 + \frac{1}{2} \frac{B^2}{\mu_0}

Applications of Electrodynamics

  1. Electric Motors and Generators: Convert electrical energy into mechanical energy and vice versa.
  2. Wireless Communication: Electromagnetic waves carry signals in radio, television, and mobile phones.
  3. Optical Systems: Light behavior is explained through electromagnetic wave theory.
  4. Power Transmission: Alternating current (AC) systems rely on electromagnetic induction.
  5. Medical Imaging: MRI machines use magnetic fields and electromagnetic waves.

Five Numerical Examples on the Applications of Electrodynamics

Example 1: Force on a Moving Charge

Problem:
A proton (𝑞 = 1.6 × 10⁻¹⁹ C) moves with a velocity of 2 × 10⁶ m/s perpendicular to a magnetic field of 0.01 T. Find the magnetic force on the proton.

Solution:
The magnetic force is given by:

𝐹=𝑞𝑣𝐵sin⁡𝜃𝐹 = 𝑞 𝑣 𝐵 \sin 𝜃

Since 𝜃 = 90°, we have sin 𝜃 = 1:

𝐹=(1.6×10−19)(2×106)(0.01)𝐹 = (1.6 × 10⁻¹⁹) (2 × 10⁶) (0.01) 𝐹=3.2×10−15 N𝐹 = 3.2 × 10⁻¹⁵ \, N

Answer:
The magnetic force is 3.2 × 10⁻¹⁵ N.

Example 2: Induced EMF in a Loop

Problem:
A circular loop of radius 0.1 m is placed in a magnetic field that increases at a rate of 0.02 T/s. Find the induced EMF.

Solution:
Faraday’s Law states:

𝜀=−dΦBdt=−AdBdt𝜀 = -\frac{dΦ_B}{dt} = – A \frac{dB}{dt}

The area of the loop is:

𝐴=πr2=π(0.1)2=0.0314 m2𝐴 = \pi r^2 = \pi (0.1)^2 = 0.0314 \, m^2 𝜀=−(0.0314)(0.02)𝜀 = – (0.0314) (0.02) 𝜀=−6.28×10−4V𝜀 = -6.28 × 10⁻⁴ V

Answer:
The induced EMF is 6.28 × 10⁻⁴ V.

Example 3: Energy Density of a Magnetic Field

Problem:
Calculate the energy density of a magnetic field with 𝐵 = 0.2 T.

Solution:
The energy density of a magnetic field is given by:

𝑢B=𝐵22μ0𝑢_B = \frac{𝐵^2}{2 \mu_0}

Substituting values:

𝑢B=(0.2)22×4π×10−7𝑢_B = \frac{(0.2)^2}{2 \times 4\pi \times 10⁻⁷} 𝑢B≈15,915 J/m3𝑢_B ≈ 15,915 \, J/m^3

Answer:
The magnetic energy density is 15,915 J/m³.

Example 4: Speed of Electromagnetic Waves in a Medium

Problem:
Find the speed of light in a medium with a relative permittivity 𝜀ᵣ = 4 and relative permeability 𝜇ᵣ = 1.

Solution:
The speed of light in a medium is given by:

𝑣=𝑐ϵrμr𝑣 = \frac{𝑐}{\sqrt{\epsilon_r \mu_r}}

Substituting values:

𝑣=3×1084×1𝑣 = \frac{3 × 10⁸}{\sqrt{4 × 1}} 𝑣=1.5×108 m/s𝑣 = 1.5 × 10⁸ \, m/s

Answer:
The speed of light in this medium is 1.5 × 10⁸ m/s.

Example 5: Poynting Vector Magnitude

Problem:
An electromagnetic wave has 𝐸 = 100 V/m. Find the magnitude of the Poynting vector.

Solution:
The Poynting vector is given by:

𝑆=𝐸2μ0c𝑆 = \frac{𝐸^2}{\mu_0 c}

Substituting values:

𝑆=(100)24π×10−7×3×108𝑆 = \frac{(100)^2}{4\pi \times 10⁻⁷ \times 3 \times 10⁸} 𝑆≈2.65 W/m2𝑆 ≈ 2.65 \, W/m^2

Answer:
The Poynting vector magnitude is 2.65 W/m².

Why Study Electrodynamics

Unification of Electricity and Magnetism

Electrodynamics describes how electric and magnetic fields interact and propagate. Students study Maxwell’s equations and the wave nature of electromagnetic fields. This unifies electric, magnetic, and optical phenomena. It is a cornerstone of modern physics.

Time-Varying Fields and Radiation

Students learn how changing electric fields create magnetic fields and vice versa. This leads to electromagnetic radiation, including light and radio waves. Understanding this interaction is key to optics, antennas, and wireless communication. It reveals how information and energy move through space.

Applications in Engineering and Technology

Electrodynamics underpins radar, telecommunications, medical imaging, and power transmission. Students explore real-world systems that rely on field interactions. This knowledge is essential for innovation in electronics and communication. It connects theory to modern technology.

Mathematical and Conceptual Mastery

Students model fields with vector calculus and partial differential equations. These methods build rigorous understanding and computational skill. They prepare students for graduate-level physics and engineering. They support precision in both theory and design.

Gateway to Advanced Electromagnetic Theory

Mastery of electrodynamics prepares students for quantum electrodynamics and particle physics. It also supports exploration of waveguides, plasmas, and optics. This topic bridges classical physics and modern theory. It equips students for research and high-level technical work.

2.65 \, \text{W/m}^2

Conclusion on Electrodynamics

Electrodynamics is a foundational pillar of physics, explaining how electric and magnetic fields interact and evolve over time. It encompasses the behavior of charges in motion, the propagation of electromagnetic waves, and the principles that underpin modern technologies such as communication systems, power generation, and medical devices. With Maxwell’s equations at its core, electrodynamics continues to drive advancements in science and engineering, impacting countless aspects of our daily lives.

Review Questions and Answers on Electrodynamics:

1. What distinguishes electrodynamics from electrostatics?
Answer: Electrodynamics studies time-varying electric and magnetic fields and their interactions, including wave propagation and induction, whereas electrostatics deals solely with stationary charges and static fields.

2. What are Maxwell’s equations and why are they central to electrodynamics?
Answer: Maxwell’s equations are four fundamental equations that describe how electric and magnetic fields are generated and interact with charges and currents. They form the theoretical backbone for understanding electromagnetic phenomena such as light propagation and electromagnetic induction.

3. How does electromagnetic induction occur and which law governs it?
Answer: Electromagnetic induction happens when a changing magnetic field within a closed loop induces an electromotive force (EMF). This phenomenon is governed by Faraday’s Law, which states that the induced EMF is proportional to the rate of change of magnetic flux through the loop.

4. What is the Lorentz force and how does it affect charged particles?
Answer: The Lorentz force is the force experienced by a charged particle moving in electric and magnetic fields, given by F = q(E + v × B). It determines the trajectory of the particle, causing circular or helical motion when the velocity is perpendicular to the magnetic field.

5. How do electric and magnetic fields work together to create electromagnetic waves?
Answer: A time-varying electric field generates a magnetic field and vice versa, leading to the self-propagation of electromagnetic waves. These waves travel through space at the speed of light, carrying energy and momentum.

6. What is the significance of the Poynting vector in electrodynamics?
Answer: The Poynting vector represents the rate of energy transfer per unit area in an electromagnetic wave. It shows the direction and magnitude of energy flow, linking the electric and magnetic fields to the energy transported by the wave.

7. How does displacement current extend Ampère’s law in Maxwell’s equations?
Answer: Displacement current is an additional term introduced by Maxwell to account for the changing electric field in regions where there is no conduction current. It ensures the continuity of current and is essential for explaining electromagnetic wave propagation.

8. What factors determine the energy of a photon in an electromagnetic wave?
Answer: The energy of a photon is determined by its frequency according to the equation E = hf, where h is Planck’s constant. Higher frequency photons carry more energy.

9. How is electromagnetic radiation applied in modern communication technologies?
Answer: Electromagnetic radiation is used to transmit information over various media. Its properties—such as frequency, wavelength, and polarization—are exploited in devices like radios, cell phones, and satellite communications to encode, transmit, and receive data.

10. Why is understanding electrodynamics crucial for innovations in renewable energy and wireless power systems?
Answer: Electrodynamics provides the principles behind energy conversion, transmission, and wireless power transfer. It enables the design of efficient solar panels, wind turbines, and wireless charging systems, which are key for advancing renewable energy technologies and sustainable power solutions.

Thought-Provoking Questions and Answers on Electrodynamics:

1. How do Maxwell’s equations unify the concepts of electricity and magnetism into a single theory?
Answer: Maxwell’s equations show that electric and magnetic fields are interdependent; a changing electric field generates a magnetic field and vice versa. This unification reveals that light itself is an electromagnetic wave, bridging the gap between the two phenomena and leading to a deeper understanding of the nature of energy.

2. In what ways might advancements in nanotechnology affect our understanding and control of electromagnetic phenomena?
Answer: Nanotechnology enables the fabrication of structures at scales comparable to the wavelength of light, leading to new optical properties and enhanced control of electromagnetic interactions. This can result in the development of metamaterials with negative refractive indices, improved sensors, and miniaturized communication devices.

3. How could the discovery of novel materials with exotic electromagnetic properties revolutionize device engineering?
Answer: Novel materials, such as topological insulators or superconductors, can exhibit unique electromagnetic behaviors like lossless conduction or robust surface states. These properties can be exploited to create highly efficient circuits, quantum computers, and advanced energy storage systems, pushing the boundaries of current technology.

4. What are the potential challenges and benefits of integrating renewable energy sources with existing electromagnetic-based power grids?
Answer: Integrating renewables requires managing variable outputs and ensuring stable power transmission. Electrodynamic principles are used to design smart grids with efficient transformers and power converters. The benefits include reduced emissions and energy sustainability, though challenges include grid stability and energy storage.

5. How does the concept of electromagnetic wave propagation influence our understanding of information transfer in modern communication systems?
Answer: Electromagnetic wave propagation allows for the transmission of signals over long distances without a physical medium. Understanding wave behavior, such as interference and diffraction, is essential for optimizing data transmission in fiber optics, wireless networks, and satellite communications.

6. What role do theoretical predictions in electrodynamics play in the development of experimental techniques in modern physics?
Answer: Theoretical predictions from electrodynamics guide experimentalists in designing apparatuses to test and measure electromagnetic phenomena. For example, predictions about light polarization and magnetic resonance have led to the development of lasers, MRI machines, and advanced spectroscopy methods.

7. How can the principles of electrodynamics be applied to improve the efficiency of wireless power transfer systems?
Answer: By optimizing the alignment, frequency, and coupling of transmitting and receiving coils based on electrodynamic principles, wireless power systems can achieve higher efficiency. Advances in resonant inductive coupling and metamaterials further enhance energy transfer over greater distances.

8. In what ways might the study of electrodynamics contribute to advancements in medical imaging and diagnostic technologies?
Answer: Electrodynamics is fundamental to techniques like MRI and X-ray imaging, where electromagnetic fields interact with tissues to produce detailed images. Further research can improve image resolution, reduce scan times, and develop non-invasive diagnostic tools that rely on controlled electromagnetic interactions.

9. How do relativistic effects alter the predictions of classical electrodynamics in high-speed systems?
Answer: At high speeds, relativistic effects such as time dilation and length contraction affect the behavior of charged particles and electromagnetic fields. This requires modifications to classical equations, leading to more accurate models for particle accelerators and astrophysical phenomena, and offering insights into the limits of classical theories.

10. What are the implications of electromagnetic compatibility (EMC) for the design of electronic devices in today’s interconnected world?
Answer: EMC ensures that electronic devices operate without interfering with one another. Designing for EMC involves careful layout, shielding, and filtering to minimize unwanted electromagnetic emissions, which is increasingly important as devices become more complex and densely integrated.

11. How might the future development of quantum electrodynamics (QED) influence our understanding of light-matter interactions?
Answer: QED provides a quantum mechanical framework for understanding the interactions between light and matter. Future advances in QED could lead to breakthroughs in controlling quantum states for computing, secure communications, and the development of new photonic devices that leverage quantum coherence and entanglement.

12. What ethical considerations might arise from the widespread deployment of advanced electromagnetic technologies in society?
Answer: As electromagnetic technologies become more pervasive, issues such as privacy, security, and health effects from prolonged exposure to electromagnetic fields must be addressed. Ethical considerations include ensuring equitable access, protecting sensitive data, and balancing technological progress with environmental and public health concerns.

13. How do Maxwell’s equations unify the phenomena of electricity and magnetism?
Answer: Maxwell’s equations reveal that electric and magnetic fields are two aspects of a single electromagnetic field. A changing electric field produces a magnetic field and vice versa, leading to the concept of electromagnetic waves and showing that light is an electromagnetic phenomenon.

14. How can advances in nanotechnology impact our control over electromagnetic phenomena?
Answer: Nanotechnology allows for the manipulation of materials at scales where quantum effects become significant. This can lead to the development of metamaterials with tailored electromagnetic properties, enabling breakthroughs in cloaking devices, highly efficient solar cells, and improved sensors.

15. In what ways might quantum electrodynamics (QED) enhance our understanding of light-matter interactions?
Answer: QED provides a quantum framework for the interactions between light and matter, accounting for phenomena such as photon emission and absorption, virtual particles, and vacuum fluctuations. It offers extremely precise predictions that are essential for technologies like lasers and quantum computing.

16. What challenges arise when attempting to harness electromagnetic induction for wireless power transfer?
Answer: Wireless power transfer faces challenges such as alignment sensitivity, efficiency losses over distance, and interference from other devices. Overcoming these requires advanced coil designs, resonant coupling techniques, and robust control systems to maximize energy transfer efficiency.

17. How does the concept of the Poynting vector inform the design of efficient antennas and communication systems?
Answer: The Poynting vector quantifies the directional energy flow of electromagnetic waves. By analyzing it, engineers can optimize antenna designs to maximize energy radiation in desired directions and improve signal strength and clarity in communication systems.

18. How might the study of electromagnetic wave propagation lead to new developments in medical imaging technologies?
Answer: Understanding wave propagation helps improve imaging techniques like MRI and CT scans. Advances in electrodynamics can lead to higher resolution images, faster scanning times, and lower power consumption, thereby enhancing diagnostic capabilities and patient outcomes.

19. How do relativistic effects influence the behavior of electromagnetic fields in high-energy environments?
Answer: At relativistic speeds, time dilation and length contraction affect how electromagnetic fields are observed. These effects require modifications to classical electrodynamic equations, impacting the design of particle accelerators and our understanding of cosmic phenomena such as pulsars and black holes.

20. What potential environmental impacts might result from widespread deployment of high-power electromagnetic systems, and how can they be mitigated?
Answer: High-power electromagnetic systems can lead to increased electromagnetic interference (EMI) and energy losses, affecting wildlife and human health. Mitigation strategies include better shielding, efficient system design, and regulatory standards to ensure minimal environmental impact.

21. How can the interplay between electric and magnetic fields be optimized to improve the efficiency of renewable energy systems?
Answer: Optimizing the coupling between electric and magnetic fields in devices like transformers, generators, and inductors can reduce energy losses and improve power conversion efficiency. Advanced materials and design techniques that enhance field interactions are crucial for next-generation renewable energy systems.

22. In what ways does electromagnetic radiation influence modern wireless communication, and what future improvements can be expected?
Answer: Electromagnetic radiation carries information in wireless communication through modulation of its amplitude, frequency, and phase. Future improvements may include the use of higher frequency bands, advanced modulation schemes, and smart antenna systems to increase data transmission rates and reduce interference.

23. How might the exploration of electromagnetic fields in space lead to breakthroughs in satellite technology and deep-space communication?
Answer: Studying electromagnetic fields in space can enhance our understanding of cosmic radiation, solar winds, and space weather, leading to improved shielding and communication systems for satellites. These advances are critical for reliable deep-space missions and global communication networks.

24. What ethical considerations arise from the development and deployment of advanced electromagnetic technologies in society?
Answer: Ethical considerations include privacy issues related to electromagnetic surveillance, the environmental impact of electromagnetic waste, and equitable access to advanced technologies. Balancing technological progress with social responsibility and environmental sustainability is crucial for ensuring that these innovations benefit society as a whole.

Numerical Problems and Solutions on Electrodynamics:

1. A circular loop of radius 0.10 m is placed in a uniform magnetic field of 0.50 T perpendicular to the loop. The magnetic field decreases uniformly to 0 T in 0.20 s. Calculate the magnitude of the induced EMF in the loop.
Solution:
  Area of the loop, A = πr² = π(0.10 m)² = 0.0314 m².
  Initial magnetic flux, Φ_i = B × A = 0.50 T × 0.0314 m² = 0.0157 Wb.
  Final flux, Φ_f = 0 Wb.
  Change in flux, ΔΦ = Φ_f − Φ_i = −0.0157 Wb.
  Induced EMF, |ε| = |ΔΦ/Δt| = 0.0157 Wb / 0.20 s = 0.0785 V.

2. A proton (charge = 1.60×10⁻¹⁹ C, mass = 1.67×10⁻²⁷ kg) moves at 2.0×10⁶ m/s perpendicular to a 0.10 T magnetic field. Calculate the radius of its circular path.
Solution:
  Radius, r = (mv) / (qB) = (1.67×10⁻²⁷ kg × 2.0×10⁶ m/s) / (1.60×10⁻¹⁹ C × 0.10 T)
    = (3.34×10⁻²¹ kg·m/s) / (1.60×10⁻²⁰ C·T) ≈ 0.209 m.

3. An electromagnetic wave in vacuum has a frequency of 5.0×10¹⁴ Hz. Calculate its wavelength.
Solution:
  Wavelength, λ = c / f = 3.00×10⁸ m/s / 5.0×10¹⁴ Hz = 6.00×10⁻⁷ m = 600 nm.

4. A plane electromagnetic wave in free space has an electric field amplitude of 200 V/m. Determine the corresponding magnetic field amplitude.
Solution:
  Magnetic field amplitude, B₀ = E₀ / c = 200 V/m / (3.00×10⁸ m/s) ≈ 6.67×10⁻⁷ T.

5. Calculate the intensity (average power per unit area) of an electromagnetic wave with an electric field amplitude of 100 V/m in free space.
Solution:
  Intensity, I = (1/2)ε₀cE₀², where ε₀ = 8.85×10⁻¹² F/m and c = 3.00×10⁸ m/s.
  I = 0.5 × 8.85×10⁻¹² F/m × 3.00×10⁸ m/s × (100 V/m)²
    = 0.5 × 8.85×10⁻¹² × 3.00×10⁸ × 10⁴
    = 13.275 W/m² (approximately).

6. In an ideal transformer, the primary coil has 1200 turns and the secondary coil has 300 turns. If the primary voltage is 240 V, what is the secondary voltage?
Solution:
  Voltage ratio, V_s/V_p = N_s/N_p = 300/1200 = 0.25.
  V_s = 240 V × 0.25 = 60 V.

7. A long straight conductor carries a current of 5 A. Calculate the magnetic field at a distance of 0.10 m from the wire. (μ₀ = 4π×10⁻⁷ T·m/A)
Solution:
  B = (μ₀I) / (2πr) = (4π×10⁻⁷ T·m/A × 5 A) / (2π × 0.10 m)
    = (20π×10⁻⁷) / (0.2π) = (20×10⁻⁷) / 0.2 = 1.0×10⁻⁵ T.

8. Two parallel wires 0.050 m apart carry currents of 10 A each in the same direction. Calculate the force per unit length between the wires.
Solution:
  Force per unit length, F/L = (μ₀I₁I₂) / (2πd) = (4π×10⁻⁷ × 10 A × 10 A) / (2π × 0.050 m)
    = (4π×10⁻⁷ × 100) / (0.1π) = (400π×10⁻⁷) / (0.1π) = 4.0×10⁻⁴ N/m.

9. A single-turn rectangular loop of area 0.02 m² rotates in a uniform magnetic field of 0.30 T. If it rotates at 10 rev/s, calculate the maximum induced EMF in the loop.
Solution:
  Angular frequency, ω = 2πf = 2π × 10 = 20π rad/s.
  Maximum induced EMF, ε_max = NABω = 1 × 0.02 m² × 0.30 T × 20π rad/s
    = 0.02 × 0.30 × 20π = 0.12π ≈ 0.377 V.

10. A solenoid has 800 turns, a length of 0.50 m, and carries a current of 3 A. Calculate the magnetic field inside the solenoid.
Solution:
  Turns per unit length, n = N/L = 800 / 0.50 = 1600 turns/m.
  Magnetic field, B = μ₀ n I = 4π×10⁻⁷ T·m/A × 1600 × 3 A
    = 4π×10⁻⁷ × 4800 = 19200π×10⁻⁷ T ≈ 6.03×10⁻³ T.

11. An electromagnetic wave in vacuum has a frequency of 2.5×10¹⁴ Hz. Calculate the energy of a single photon.
Solution:
  Photon energy, E = hf, where h = 6.626×10⁻³⁴ J·s.
  E = 6.626×10⁻³⁴ J·s × 2.5×10¹⁴ Hz ≈ 1.66×10⁻¹⁹ J.

12. A rectangular loop of wire (0.3 m by 0.2 m) is rotated from a position perpendicular to a 0.40 T magnetic field to a position parallel to the field in 0.50 s. Calculate the average induced EMF in the loop.
Solution:
  Initial magnetic flux, Φ_i = B × A = 0.40 T × (0.3 m × 0.2 m) = 0.40 × 0.06 = 0.024 Wb.
  Final flux, Φ_f = 0 Wb (when parallel, flux is zero).
  Change in flux, ΔΦ = 0 − 0.024 Wb = −0.024 Wb.
  Average induced EMF, |ε_avg| = |ΔΦ/Δt| = 0.024 Wb / 0.50 s = 0.048 V.