Electromagnetic Waves

Explanation with Reasons

(i) Electromagnetic Waves Differ Considerably in Their Mode of Interaction with Matter Electromagnetic waves cover a wide spectrum, ranging from radio waves to gamma rays. The way they interact with matter depends on their frequency and wavelength:
1. Low-frequency waves (e.g., radio waves) can pass through materials like air, glass, and other non-metallic substances because their energy is not high enough to excite electrons or cause significant atomic or molecular transitions.
2. High-frequency waves (e.g., UV rays, X-rays, and gamma rays) have enough energy to ionize atoms or molecules, leading to chemical reactions or causing ionization, which is harmful and potentially dangerous. Therefore, electromagnetic waves interact with matter differently depending on their energy content, which varies with frequency and wavelength.

(ii) Food Items to Be Heated in Microwave Ovens Must Contain Water Microwave ovens work by emitting electromagnetic waves with a frequency typically around 2.45 GHz. These microwaves are absorbed by water molecules in food. The microwaves cause the water molecules to rotate rapidly, generating heat through friction. Since water has a high dielectric constant, it absorbs microwaves effectively, heating the food. Foods without water, like dry bread or certain snacks, do not absorb microwaves efficiently, so they do not heat up well. Therefore, food items with water are best suited for microwave heating.

(iii) Welders Wear Face Mask with Glasses During Welding During welding, the intense light and ultraviolet radiation emitted can cause severe damage to the eyes, resulting in conditions like welder’s flash or arc eye. These conditions are similar to sunburn but affect the eyes, causing pain, inflammation, and even long-term damage. The face mask with protective glasses worn by welders serves to block harmful UV rays and visible light from the welding arc, protecting their eyes from injury. The lenses in the welding glasses are specifically designed to filter out harmful radiation while allowing sufficient light for visibility, ensuring both safety and clarity during the welding process.

Understanding Displacement Current and Continuity in a Charging Capacitor

1: Definition of Displacement Current
- The displacement current ( ) is a term introduced by James Clerk Maxwell to explain how a changing electric field can produce a magnetic field, similar to a conduction current.
- It is given by: where:
- = Permittivity of free space,
- = Electric flux through a given surface.
2: Continuity of Current in a Charging Capacitor
- Consider a capacitor connected to a DC source.
- During charging, conduction current flows in the wires.
- Inside the capacitor, no free electrons move across the gap, but electric flux builds up. - Maxwell’s Equation states that displacement current maintains continuity, ensuring a continuous magnetic field around the circuit.
- Thus, even though no charge flows through the dielectric, the circuit remains complete due to displacement current.
3: Conclusion
- Displacement current is required to modify Ampère’s Law and explain how a capacitor circuit remains continuous even in the absence of conduction current in the capacitor gap.

Electromagnetic Waves and Their Applications

Explanation:
- FM Radio Broadcast: Uses radio waves in the meter range, allowing transmission over long distances.
- Detection of Bone Fractures: Uses X-rays as they can penetrate soft tissues but are absorbed by bones, creating images.
- Treatment of Muscular Strain: Uses infrared rays to provide deep tissue heating and pain relief.

X-rays and ultraviolet (UV) rays both belong to the electromagnetic spectrum, but X-rays have a much smaller wavelength and hence higher energy than UV rays. The energy of a photon is given by: where:
is energy,
is Planck’s constant,
is frequency,
is the speed of light,
is the wavelength.

Since X-rays have a smaller wavelength ( ) than UV rays, they carry more energy per photon. This higher energy enables them to penetrate deeper into tissues, making them more harmful to living cells. Thus, the correct answer is (2).

We are tasked with estimating the ratio of the shortest wavelength of radio waves to the longest wavelength of gamma waves. Let's first define the given quantities:

1. Shortest Wavelength of Radio Waves:

The shortest wavelength of radio waves is typically in the range of (10 cm). Thus, we can take:

2. Longest Wavelength of Gamma Waves:

The longest wavelength of gamma rays is typically on the order of (1 picometer). Thus, we can take:

3. Ratio of Wavelengths:

The ratio of the shortest wavelength of radio waves to the longest wavelength of gamma waves is given by:

4. Conclusion:

The ratio of the shortest wavelength of radio waves to the longest wavelength of gamma waves is , or 100 billion.

Validity of Kirchhoff's First Law at Capacitor Plates

Problem Statement:

A parallel plate capacitor is charged by an AC source. Show that the sum of conduction current and displacement current has the same value at all points in the circuit.

Explanation:

1. What is Conduction Current and Displacement Current?

• Conduction Current : The usual electric current due to flow of charges in a conductor (e.g., wires).
• Displacement Current : A current-like term introduced by Maxwell in the dielectric (non-conducting) region between capacitor plates, given by:
  , where is the electric flux.

2. AC Source with Capacitor:

In an AC circuit, the current continuously changes direction. Charges accumulate and deplete on the capacitor plates, creating a time-varying electric field between them. This changing field gives rise to a displacement current in the dielectric.

3. Continuity of Current:

Maxwell showed that:

This equality ensures that the current appears continuous throughout the entire circuit, including the space between the capacitor plates where no actual charge carriers move.

Answer:

Yes, Kirchhoff’s first law (junction rule) is valid at each plate of the capacitor, because the sum of the conduction current and the displacement current is the same at all points in the circuit.

Reason: The displacement current ensures there is no accumulation of charge at any point in the circuit. Therefore, current continuity is maintained, and the junction rule:

holds true even at the surfaces of the capacitor plates.

Conclusion:

The concept of displacement current bridges the gap in the dielectric region of a capacitor, thereby upholding Kirchhoff’s current law universally, even in time-varying (AC) circuits.

In a parallel plate capacitor, the conduction current is the current flowing through the plates, and the displacement current is the current that is related to the changing electric field between the plates. The total current is the sum of both: For a capacitor, the conduction current is given by: where is the charge on the capacitor. The displacement current is related to the rate of change of the electric field between the plates: where is the area of the plates and is the electric field between the plates. Since the displacement current is equivalent to the conduction current in terms of charge flow, we have: Thus, the sum of the conduction and displacement currents is the same at all points in the circuit.

We are given an electromagnetic wave traveling in the vertically upward direction, and at a certain instant, its electric field vector points in the west direction. We are tasked with determining the direction of the magnetic field vector at that instant.

1. Understanding the Relationship Between Electric and Magnetic Fields in an Electromagnetic Wave:

In an electromagnetic wave, the electric field ( ) and magnetic field ( ) are always perpendicular to each other and also perpendicular to the direction of propagation of the wave. This means the three vectors—electric field, magnetic field, and the direction of wave propagation—form a right-handed coordinate system.

2. Direction of Propagation and Fields:

Let's define the directions using a coordinate system:

• The wave is traveling in the vertically upward direction, which corresponds to the -axis in a typical 3D coordinate system.
• The electric field vector ( ) points in the west direction, which corresponds to the -axis (since west is usually defined as negative -direction in a standard geographic coordinate system).
• According to the right-hand rule, the magnetic field ( ) must be perpendicular to both the electric field vector ( ) and the direction of propagation of the wave (the -axis).

3. Applying the Right-Hand Rule:

To find the direction of the magnetic field vector ( ), we use the right-hand rule, which states that if the thumb of your right hand points in the direction of wave propagation (upward, or -axis), and the index finger points in the direction of the electric field ( -axis, or west), then the middle finger will point in the direction of the magnetic field.

Using the right-hand rule, with the electric field pointing west (along ) and the wave propagating vertically upward (along ), the magnetic field vector will point in the north direction, or along the -axis.

4. Conclusion:

At the instant when the electric field vector points in the west direction, the magnetic field vector points in the north direction.

An electromagnetic wave is produced by the acceleration of charged particles, typically electrons, in a conductor or in free space. The process involves the creation of oscillating electric and magnetic fields that propagate through space. Let's break down the production of electromagnetic waves step by step:

1. Accelerating Charge Creates Electric and Magnetic Fields:

The fundamental process behind the production of an electromagnetic wave is the acceleration of a charged particle. When a charged particle, such as an electron, is accelerated (either by an alternating current or by an external force), it creates disturbances in both the electric and magnetic fields around it.

• When the charge accelerates, it produces a changing electric field around it. A changing electric field generates a magnetic field, and similarly, a changing magnetic field generates an electric field. This interplay between the electric and magnetic fields creates the electromagnetic wave.

2. Oscillating Electric and Magnetic Fields:

The accelerated charge causes the electric field ( ) and magnetic field ( ) to oscillate. These fields are perpendicular to each other and to the direction of wave propagation. Here's how it works:

• The electric field oscillates in one direction (say, along the -axis), and the magnetic field oscillates in a direction perpendicular to both the electric field and the direction of wave travel (say, along the -axis).
• These oscillating electric and magnetic fields propagate outward from the source of the disturbance, forming the electromagnetic wave. The wave travels at the speed of light in a vacuum ( ).

3. Role of Electromagnetic Wave Generation in Different Scenarios:

Electromagnetic waves are generated in various scenarios, such as:

• Antennas: In radio transmission, antennas produce electromagnetic waves by accelerating electrons back and forth, which creates oscillating electric and magnetic fields that radiate away from the antenna as electromagnetic waves.
• Light Sources: In light sources like incandescent bulbs or lasers, electrons in atoms are accelerated when they absorb energy. This causes them to transition between energy levels and emit photons, which are quantized electromagnetic waves (light).
• Magnetic Induction: In changing magnetic fields, according to Faraday's law of induction, a changing magnetic field induces an electric field, which can propagate as an electromagnetic wave. This is the basis for the generation of electromagnetic waves in electric circuits, such as in transformers.

4. Conclusion:

An electromagnetic wave is produced when charged particles accelerate, creating oscillating electric and magnetic fields. These fields propagate through space in the form of an electromagnetic wave, traveling at the speed of light. The interaction between the electric and magnetic fields is what sustains the wave as it moves through space.

To understand the change in wave characteristics from microwaves to X-rays in the electromagnetic spectrum, we need to consider the general properties of electromagnetic waves:

• Speed: Electromagnetic waves travel at the same speed in a vacuum, which is approximately c = 3 × 10⁸ m/s. Therefore, the speed remains constant across the spectrum.
• Wavelength and Frequency: The relationship between speed (c), frequency (f), and wavelength (λ) is given by the equation c = λf. When the speed is constant, any change in wavelength results in an inverse change in frequency.

From microwaves to X-rays:

• Wavelength: Generally decreases. Microwaves have longer wavelengths compared to X-rays.
• Frequency: Increases. Since their wavelengths decrease, their frequencies must increase to maintain the constant speed, according to the formula.

Thus, the correct change in wave characteristics from microwaves to X-rays is: "Speed remains same, Wavelength decreases, Frequency increases." This is captured in the correct answer given.