1. Electromagnetic Waves Physics 4 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

2. Maxwell’s Equations Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Maxwell’s equations summarize the relationships between electric and magnetic fields. A major consequence of these equations is that an accelerating charge will produce electromagnetic radiation.

3. Electromagnetic (EM) waves can be produced by atomic transitions (more on this later), or by an alternating current in a wire. As the charges in the wire oscillate back and forth, the electric field around them oscillates as well, in turn producing an oscillating magnetic field. We have a right-hand-rule for plane EM waves: 1) Point the fingers of your right hand in the direction of the E-field 2) Curl them toward the B-field. 3) Stick out your thumb - it points in the direction of propagation. Electromagnetic Waves Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Click here for an EM wave animation

4. Like any other wave, we know the relationship between the wavelength and frequency, and the speed of propagation of the wave: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

5. Like any other wave, we know the relationship between the wavelength and frequency, and the speed of propagation of the wave: In the case of EM waves, it turns out that the wave speed is the speed of light. So our formula for EM waves (in vacuum) is: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

6. Like any other wave, we know the relationship between the wavelength and frequency, and the speed of propagation of the wave: In the case of EM waves, it turns out that the wave speed is the speed of light. So our formula for EM waves (in vacuum) is: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB The speed of light is also related to the strengths of the Electric and Magnetic fields. E=cB (in standard metric units)

7. The continuum of various wavelengths and frequencies for EM waves is called the Electromagnetic Spectrum Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

8. The continuum of various wavelengths and frequencies for EM waves is called the Electromagnetic Spectrum Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Find the frequency of blue light with a wavelength of 460 nm.

9. The continuum of various wavelengths and frequencies for EM waves is called the Electromagnetic Spectrum Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Find the frequency of blue light with a wavelength of 460 nm.

10. The continuum of various wavelengths and frequencies for EM waves is called the Electromagnetic Spectrum Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB A cell phone transmits at a frequency of 1.25x10 8 Hz. What is the wavelength of this EM wave?

11. The continuum of various wavelengths and frequencies for EM waves is called the Electromagnetic Spectrum Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB A cell phone transmits at a frequency of 1.25x10 8 Hz. What is the wavelength of this EM wave?

12. Field of a Sinusoidal Wave Electromagnetic waves must satisfy the WAVE EQUATION: In the case of EM waves, both the electric and magnetic fields need to satisfy this equation. Solving this equation yields formulas for the E and B fields. In particular, here are formulas for the E and B fields associated with a sinusoidal EM plane wave propagating in the +x-direction: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Notice that these fields are perpendicular to each other, as well as the propagation direction. A right hand rule comes in handy to remember the directions.

13. Field of a Sinusoidal Wave Electromagnetic waves must satisfy the WAVE EQUATION: In the case of EM waves, both the electric and magnetic fields need to satisfy this equation. Solving this equation yields formulas for the E and B fields. In particular, here are formulas for the E and B fields associated with a sinusoidal EM plane wave propagating in the +x-direction: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Notice that these fields are perpendicular to each other, as well as the propagation direction. A right hand rule comes in handy to remember the directions. Example: A sinusoidal EM wave of frequency 6.10x10 14 Hz travels in vacuum in the +z-direction. The B-field is parallel to the y-axis and has amplitude 5.80x10 -4 T. Write the equations for the E and B fields.

14. Field of a Sinusoidal Wave Electromagnetic waves must satisfy the WAVE EQUATION: In the case of EM waves, both the electric and magnetic fields need to satisfy this equation. Solving this equation yields formulas for the E and B fields. In particular, here are formulas for the E and B fields associated with a sinusoidal EM plane wave propagating in the +x-direction: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Notice that these fields are perpendicular to each other, as well as the propagation direction. A right hand rule comes in handy to remember the directions. Example: A sinusoidal EM wave of frequency 6.10x10 14 Hz travels in vacuum in the +z-direction. The B-field is parallel to the y-axis and has amplitude 5.80x10 -4 T. Write the equations for the E and B fields.

15. EM Waves in matter So far we have assumed that electromagnetic waves propagated through empty space. If they travel through a transparent material medium (glass, air, water, etc.) the speed of propagation changes. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB This is the speed in vacuum This is the speed in a material medium with dielectric constant * K and relative permeability K m For most materials K m is close to one, so we can effectively ignore it and get n is called the index of refraction for the medium Since K>1, the speed of an EM wave in a material medium is always less than c. *K is not technically a constant – when rapidly oscillating fields are present the value is usually smaller than with constant fields, so the value of K is dependent on the frequency of the EM wave.

16. Energy and momentum in EM Waves Electromagnetic waves transport energy. The energy associated with a wave is stored in the oscillating electric and magnetic fields. We will find out later that the frequency of the wave determines the amount of energy that it carries. Since the EM wave is in 3-D, we need to measure the energy density (energy per unit volume). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB This is the energy per unit volume

17. Energy and momentum in EM Waves Electromagnetic waves transport energy. The energy associated with a wave is stored in the oscillating electric and magnetic fields. We will find out later that the frequency of the wave determines the amount of energy that it carries. Since the EM wave is in 3-D, we need to measure the energy density (energy per unit volume). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB This is the energy per unit volume The Poynting vector describes the energy flow rate. This vector usually oscillates rapidly, so it makes sense to talk about the average value, which turns out to be the INTENSITY of the radiation, with units W/m 2. For a sinusoidal wave in vacuum we can write this in several forms:

18. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example: High-Energy Cancer Treatment Scientists are working on a technique to kill cancer cells by zapping them with ultrahigh- energy pulses of light that last for an extremely short amount of time. These short pulses scramble the interior of a cell without causing it to explode, as long pulses do. We can model a typical such cell as a disk 5.0 µm in diameter, with the pulse lasting for 4.0 ns with a power of 2.0x10 12 W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. a) How much energy is given to the cell during this pulse? b) What is the intensity (in W/m 2 ) delivered to the cell? c) What are the maximum values of the electric and magnetic fields in the pulse?

19. Recall that power is energy/time. So 2.0x10 12 W is 2.0x10 12 Joules/sec. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB This is the total energy, which is spread out over 100 cells, so the energy for each individual cell is 80 Joules. Example: High-Energy Cancer Treatment Scientists are working on a technique to kill cancer cells by zapping them with ultrahigh- energy pulses of light that last for an extremely short amount of time. These short pulses scramble the interior of a cell without causing it to explode, as long pulses do. We can model a typical such cell as a disk 5.0 µm in diameter, with the pulse lasting for 4.0 ns with a power of 2.0x10 12 W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. a) How much energy is given to the cell during this pulse? b) What is the intensity (in W/m 2 ) delivered to the cell? c) What are the maximum values of the electric and magnetic fields in the pulse?

20. To get intensity, we need to divide power/area. The area for a cell is just the area of a circle: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example: High-Energy Cancer Treatment Scientists are working on a technique to kill cancer cells by zapping them with ultrahigh- energy pulses of light that last for an extremely short amount of time. These short pulses scramble the interior of a cell without causing it to explode, as long pulses do. We can model a typical such cell as a disk 5.0 µm in diameter, with the pulse lasting for 4.0 ns with a power of 2.0x10 12 W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. a) How much energy is given to the cell during this pulse? b) What is the intensity (in W/m 2 ) delivered to the cell? c) What are the maximum values of the electric and magnetic fields in the pulse?

21. To get intensity, we need to divide power/area. The area for a cell is just the area of a circle: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Now divide to get intensity: This is the total area of all 100 cells. Example: High-Energy Cancer Treatment Scientists are working on a technique to kill cancer cells by zapping them with ultrahigh- energy pulses of light that last for an extremely short amount of time. These short pulses scramble the interior of a cell without causing it to explode, as long pulses do. We can model a typical such cell as a disk 5.0 µm in diameter, with the pulse lasting for 4.0 ns with a power of 2.0x10 12 W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. a) How much energy is given to the cell during this pulse? b) What is the intensity (in W/m 2 ) delivered to the cell? c) What are the maximum values of the electric and magnetic fields in the pulse?

22. To get the field strengths, recall our intensity formula: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example: High-Energy Cancer Treatment Scientists are working on a technique to kill cancer cells by zapping them with ultrahigh- energy pulses of light that last for an extremely short amount of time. These short pulses scramble the interior of a cell without causing it to explode, as long pulses do. We can model a typical such cell as a disk 5.0 µm in diameter, with the pulse lasting for 4.0 ns with a power of 2.0x10 12 W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. a) How much energy is given to the cell during this pulse? b) What is the intensity (in W/m 2 ) delivered to the cell? c) What are the maximum values of the electric and magnetic fields in the pulse?

23. EM waves also carry momentum. This means that a ray of light can actually exert a force. To get the pressure exerted by a sinusoidal EM wave, just divide the intensity by the speed of light. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB This is the same as the total energy absorbed by the surface. If the energy is reflected, the pressure is doubled. Energy and momentum in EM Waves

24. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example: Solar Sails Suppose a spacecraft with a mass of 25,000 kg has a solar sail made of perfectly reflective aluminized film with an area of 2.59x10 6 m. If the spacecraft is launched into earth orbit and then deploys its sail at right angles to the sunlight, what is the acceleration due to sunlight? Assume that at the earth’s distance from the sun, the intensity of sunlight is 1410 W/m 2.

25. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Recall that Pressure = Force/Area. We can use this and F=ma to get our formula: Since the sunlight reflects from our solar sail we should double the given pressure. Example: Solar Sails Suppose a spacecraft with a mass of 25,000 kg has a solar sail made of perfectly reflective aluminized film with an area of 2.59x10 6 m. If the spacecraft is launched into earth orbit and then deploys its sail at right angles to the sunlight, what is the acceleration due to sunlight? Assume that at the earth’s distance from the sun, the intensity of sunlight is 1410 W/m 2.

26. When EM waves are reflected we can have a superposition of waves traveling in opposite directions, forming a STANDING WAVE. After combining the formulas for the opposite-directed waves, and applying a bit of trigonometry, we arrive at formulas for the E and B fields of a standing EM wave. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Standing EM Waves We can find the positions where these fields go to zero (at all times t). These are called the NODAL PLANES: For the E-field we need sin(kx)=0, which leads to the following locations: Similarly for the B-field we need cos(kx)=0, which gives:

27. If we have 2 reflecting surfaces parallel to each other we can “trap” a standing EM wave in a box, just like having a standing wave on a stretched string. The formulas are even the same: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Standing EM Waves These formulas give the wavelengths and frequencies for standing waves that will “fit” in a box of length L