Electromagnetic Waves (EM Waves)

Electromagnetic waves are waves that can travel through the vacuum of space as well as through various mediums. They are formed when an electric field and a magnetic field interact and oscillate together at right angles to each other. These waves transport energy through space.

Characteristics of Electromagnetic Waves:

1. Transverse Nature:

   • Electromagnetic waves are transverse waves, meaning the oscillations of the electric and magnetic fields are perpendicular to the direction of wave propagation.

2. Speed of Light:

   • In a vacuum, the speed of an electromagnetic wave is approximately 3 x 10^8 meters per second (the speed of light, c).

3. Wavelength and Frequency:

   • The wavelength ($\lambda$) is the distance between two consecutive crests or troughs of a wave.
   • The frequency (f) is the number of complete wave cycles that pass a point per second.
   • The relationship between wavelength, frequency, and the speed of light is given by: $$c = \lambda \cdot f$$where:
     • $c$ is the speed of light,
     • $\lambda$ is the wavelength,
     • $f$ is the frequency.

4. Energy:

   • The energy of an electromagnetic wave is directly proportional to its frequency: $$E = h \cdot f$$where:
     • $E$ is energy,
     • $h$ is Planck’s constant ($6.626 \times 10^{-34} J \cdot s$),
     • $f$ is the frequency of the wave.

The Electromagnetic Spectrum:

Electromagnetic waves vary in wavelength and frequency. This range is known as the electromagnetic spectrum, which includes different types of waves with distinct properties.

1. Radio Waves:

   • Wavelength: $> 10^2 \, m$
     
   • Frequency: $< 10^9 \, Hz$
     
   • Uses: Communication, broadcasting, and radar.

2. Microwaves:

   • Wavelength: $10^{-2} \, m$ to $10^{-3} \, m$
     
   • Frequency: $10^9 \, Hz$ to $10^{12} \, Hz$
     
   • Uses: Cooking (microwave ovens), satellite communication, and radar.

3. Infrared (IR):

   • Wavelength: $10^{-5} \, m$ to $10^{-6} \, m$
     
   • Frequency: $10^{12} \, Hz$ to $10^{14} \, Hz$
     
   • Uses: Thermal imaging, remote controls, and heat sensing.

4. Visible Light:

   • Wavelength: $4 \times 10^{-7} \, m$ to $7 \times 10^{-7} \, m$
     
   • Frequency: $4 \times 10^{14} \, Hz$ to $7 \times 10^{14} \, Hz$
     
   • Uses: Human vision.

5. Ultraviolet (UV):

   • Wavelength: ${10}^{-8}\mathrm{m\; to }$$10^{-9} \, m$
     
   • Frequency: $10^{15} \, Hz$ to $10^{16} \, Hz$
     
   • Uses: Sterilization, blacklights, and skin tanning.

6. X-rays:

   • Wavelength: $10^{-11} \, m$ to $10^{-8} \, m$
     
   • Frequency: $10^{16} \, Hz$ to ${10}^{20}Hz$
   • Uses: Medical imaging and cancer treatment.

7. Gamma Rays:

   • Wavelength: $< 10^{-12} \, m$
     
   • Frequency: $> 10^{20} \, Hz$
     
   • Uses: Cancer treatment and sterilization.

Properties of Electromagnetic Waves:

1. Reflection:

   • EM waves can reflect off surfaces, obeying the laws of reflection (angle of incidence = angle of reflection).

2. Refraction:

   • When electromagnetic waves pass from one medium to another, they change speed and direction. This is known as refraction.

3. Diffraction:

   • EM waves can bend around obstacles and spread out through small openings, a phenomenon known as diffraction.

4. Interference:

   • When two or more electromagnetic waves meet, they can interfere with each other, leading to constructive or destructive interference.

Speed of EM Waves in Different Mediums:

1. In a Vacuum:

   • As mentioned, the speed of light (EM waves) in a vacuum is $c = 3.00 \times 10^8 \, \text{m/s}$.

2. In Other Media:

   • When electromagnetic waves travel through other materials (like air, water, glass, etc.), their speed decreases due to the refractive index of the medium.
   • The refractive index (n) of a medium is given by: $$n = \frac{c}{v}$$ where:
     • $c$ is the speed of light in a vacuum,
     • $v$ is the speed of light in the medium.

   The speed of an EM wave in a medium can be calculated as:

   $$v = \frac{c}{n}$$

   For example:

   • In air, the refractive index is close to 1, so the speed of EM waves is nearly the same as in a vacuum.
   • In water, the refractive index is around 1.33, so the speed of EM waves in water would be slower: $$v = \frac{3.00 \times 10^8}{1.33} \approx 2.26 \times 10^8 \, \text{m/s}$$
     

3. In Materials like Glass or Diamond:

   • In glass, the refractive index typically ranges from 1.5 to 1.6, so the speed of light would be slower than in air.
   • In diamond, the refractive index is about 2.42, leading to even slower speeds.

Solved Problems

1. The speed of light in water is approximately $2.25 \times 10^8 \, \text{m/s}$. Calculate the refractive index of water.

Solution:
We know the relationship between the speed of light in a vacuum ($c$) and the speed of light in a medium ($v$):

$$n = \frac{c}{v}$$

Where:

• $c=3.00\times {10}^8\text{m/s (speed of light in vacuum),}$
• $v = 2.25 \times 10^8 \, \text{m/s}$ (speed of light in water).

Now, substituting the values into the equation:

$$n = \frac{3.00 \times 10^8}{2.25 \times 10^8}$$ $$n=1.33$$

The refractive index of water is $n = 1.33$.

2. A radio wave has a frequency of $9.0 \times 10^6 \, \text{Hz}$. What is its wavelength in a vacuum?

Solution:
We can use the relationship between wavelength, frequency, and the speed of light:

$$c=\lambda \cdot f$$

Where:

• $c = 3.00 \times 10^8 \, \text{m/s}$ (speed of light in a vacuum),
• $\lambda$ is the wavelength,
• $f = 9.0 \times 10^6 \, \text{Hz}$ (frequency).

Rearrange the formula to solve for wavelength ($\lambda$):

$$\lambda =\frac{c}{\mathrm{f}}$$

Now, substitute the values:

$$\lambda = \frac{3.00 \times 10^8}{9.0 \times 10^6}$$ $$\lambda = 33.33 \, \text{m}$$

The wavelength of the radio wave is $33.33 \, \text{m}$.

3. The speed of light in glass is $2.0 \times 10^8 \, \text{m/s}$. If the refractive index of glass is 1.5, calculate the speed of light in a vacuum.

Solution:
We know the formula for the refractive index:

$$n=\frac{c}{\mathrm{v}}$$

Where:

• $n = 1.5$ (refractive index of glass),
• $v = 2.0 \times 10^8 \, \text{m/s}$ (speed of light in glass),
• $c$ is the speed of light in a vacuum.

Rearrange the formula to solve for $c$:

$$c=n\cdot v$$

Now, substitute the values:

$$c = 1.5 \times 2.0 \times 10^8$$
$$c = 3.0 \times 10^8 \, \text{m/s}$$

The speed of light in a vacuum is $3.0 \times 10^8 \, \text{m/s}$, which matches the known speed of light.

4. An electromagnetic wave has a frequency of $5.0 \times 10^{14} \, \text{Hz}$. What is its energy?

Solution:
The energy of an EM wave can be calculated using the formula:

$$E = h \cdot f$$

Where:

• $E$ is the energy of the wave,
• $h = 6.626 \times 10^{-34} \, \text{J} \cdot \text{s}$ (Planck’s constant),
• $f = 5.0 \times 10^{14} \, \text{Hz}$ (frequency).

Now, substitute the values:

$$E = (6.626 \times 10^{-34}) \times (5.0 \times 10^{14})$$
$$E=3.313\times {10}^{-19}\text{J}$$

The energy of the electromagnetic wave is $3.313 \times 10^{-19} \, \text{J}$.

Problems

1. A light wave travels through a material with a refractive index of $1.8$ and a speed of $2.4 \times 10^8 \, \text{m/s}$. Calculate the speed of light in a vacuum.

2. A light wave has a wavelength of $600 \, \text{nm}$ (nanometers) and travels through air. Calculate the frequency of the light wave.

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