Electromagnetic Waves

Transcription

1 Electroagnetic Waves Physics 4

2 Maxwell s Equations Maxwell s equations suarize the relationships between electric and agnetic fields. A ajor consequence of these equations is that an accelerating charge will produce electroagnetic radiation. E d A = Q encl ε 0 B d A = 0 Gauss s Law for E Gauss s Law for B B d l = μ 0 i c + ε 0 dφ E dt encl Apere s Law E d l = dφ B dt Faraday s Law

3 Electroagnetic (EM) waves can be produced by atoic transitions (ore 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 the oscillates as well, in turn producing an oscillating agnetic 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 ) Curl the toward the B-field. Electroagnetic Waves 3) Stick out your thub - it points in the direction of propagation. Click here for an EM wave aniation

4 Like any other wave, we know the relationship between the wavelength and frequency, and the speed of propagation of the wave: v wave f

5 Like any other wave, we know the relationship between the wavelength and frequency, and the speed of propagation of the wave: v wave f In the case of EM waves, it turns out that the wave speed is the speed of light. So our forula for EM waves (in vacuu) is: c f 1 ; c s

6 Like any other wave, we know the relationship between the wavelength and frequency, and the speed of propagation of the wave: v wave f In the case of EM waves, it turns out that the wave speed is the speed of light. So our forula for EM waves (in vacuu) is: c f 1 ; c s The speed of light is also related to the strengths of the Electric and Magnetic fields. E=cB (in standard etric units)

7 The continuu of various wavelengths and frequencies for EM waves is called the Electroagnetic Spectru

8 The continuu of various wavelengths and frequencies for EM waves is called the Electroagnetic Spectru Find the frequency of blue light with a wavelength of 460 n.

9 The continuu of various wavelengths and frequencies for EM waves is called the Electroagnetic Spectru Find the frequency of blue light with a wavelength of 460 n. c f f c s Hz

10 The continuu of various wavelengths and frequencies for EM waves is called the Electroagnetic Spectru A cell phone transits at a frequency of 1.5x10 8 Hz. What is the wavelength of this EM wave?

11 The continuu of various wavelengths and frequencies for EM waves is called the Electroagnetic Spectru A cell phone transits at a frequency of 1.5x10 8 Hz. What is the wavelength of this EM wave? c f c f s Hz.4

12 Field of a Sinusoidal Wave Electroagnetic waves ust satisfy the WAVE EQUATION: In the case of EM waves, both the electric and agnetic fields need to satisfy this equation. Solving this equation yields forulas for the E and B fields. In particular, here are forulas for the E and B fields associated with a sinusoidal EM plane wave propagating in the +x-direction: y(x, t) x = 1 v y(x, t) t E x, t = E ax cos(kx ωt) j B x, t = B ax cos(kx ωt) k k = wave nuber = π λ ω = angular frequency = πf Notice that these fields are perpendicular to each other, as well as the propagation direction. A right hand rule coes in handy to reeber the directions.

13 Field of a Sinusoidal Wave Electroagnetic waves ust satisfy the WAVE EQUATION: In the case of EM waves, both the electric and agnetic fields need to satisfy this equation. Solving this equation yields forulas for the E and B fields. In particular, here are forulas for the E and B fields associated with a sinusoidal EM plane wave propagating in the +x-direction: y(x, t) x = 1 v y(x, t) t E x, t = E ax cos(kx ωt) j B x, t = B ax cos(kx ωt) k k = wave nuber = π λ ω = angular frequency = πf Notice that these fields are perpendicular to each other, as well as the propagation direction. A right hand rule coes in handy to reeber the directions. Exaple: A sinusoidal EM wave of frequency 6.10x10 14 Hz travels in vacuu in the +z-direction. The B-field is parallel to the y-axis and has aplitude 5.80x10-4 T. Write the equations for the E and B fields.

14 Field of a Sinusoidal Wave Electroagnetic waves ust satisfy the WAVE EQUATION: In the case of EM waves, both the electric and agnetic fields need to satisfy this equation. Solving this equation yields forulas for the E and B fields. In particular, here are forulas for the E and B fields associated with a sinusoidal EM plane wave propagating in the +x-direction: y(x, t) x = 1 v y(x, t) t E x, t = E ax cos(kx ωt) j B x, t = B ax cos(kx ωt) k k = wave nuber = π λ ω = angular frequency = πf Notice that these fields are perpendicular to each other, as well as the propagation direction. A right hand rule coes in handy to reeber the directions. Exaple: A sinusoidal EM wave of frequency 6.10x10 14 Hz travels in vacuu in the +z-direction. The B-field is parallel to the y-axis and has aplitude 5.80x10-4 T. Write the equations for the E and B fields. E z, t = E ax cos(kz ωt) i B z, t = B ax cos(kz ωt) j E ax = cb ax = s ω = π Hz = rad s k = ω c = rad s s T = V = rad

15 EM Waves in atter So far we have assued that electroagnetic waves propagated through epty space. If they travel through a transparent aterial ediu (glass, air, water, etc.) the speed of propagation changes. c = 1 ε 0 μ 0 This is the speed in vacuu v = 1 εμ = 1 Kε 0 K μ 0 = c KK = c n This is the speed in a aterial ediu with dielectric constant * K and relative pereability K For ost aterials K is close to one, so we can effectively ignore it and get n = KK K n is called the index of refraction for the ediu Since K>1, the speed of an EM wave in a aterial ediu is always less than c. *K is not technically a constant when rapidly oscillating fields are present the value is usually saller than with constant fields, so the value of K is dependent on the frequency of the EM wave.

16 Energy and oentu in EM Waves Electroagnetic waves transport energy. The energy associated with a wave is stored in the oscillating electric and agnetic fields. We will find out later that the frequency of the wave deterines the aount of energy that it carries. Since the EM wave is in 3-D, we need to easure the energy density (energy per unit volue). u = 1 ε 0E + 1 μ 0 B = ε 0 E This is the energy per unit volue

17 Energy and oentu in EM Waves Electroagnetic waves transport energy. The energy associated with a wave is stored in the oscillating electric and agnetic fields. We will find out later that the frequency of the wave deterines the aount of energy that it carries. Since the EM wave is in 3-D, we need to easure the energy density (energy per unit volue). u = 1 ε 0E + 1 μ 0 B = ε 0 E This is the energy per unit volue The Poynting vector describes the energy flow rate. S = 1 μ 0 E B This vector usually oscillates rapidly, so it akes sense to talk about the average value, which turns out to be the INTENSITY of the radiation, with units W/. For a sinusoidal wave in vacuu we can write this in several fors: I = S av = E axb ax μ 0 = E ax μ 0 c = 1 ε 0cE ax

18 Exaple: High-Energy Cancer Treatent Scientists are working on a technique to kill cancer cells by zapping the with ultrahighenergy pulses of light that last for an extreely short aount of tie. These short pulses scrable the interior of a cell without causing it to explode, as long pulses do. We can odel a typical such cell as a disk 5.0 µ in diaeter, with the pulse lasting for 4.0 ns with a power of.0x10 1 W. We shall assue that the energy is spread uniforly over the faces of 100 cells for each pulse. a) How uch energy is given to the cell during this pulse? b) What is the intensity (in W/ ) delivered to the cell? c) What are the axiu values of the electric and agnetic fields in the pulse?

19 Exaple: High-Energy Cancer Treatent Scientists are working on a technique to kill cancer cells by zapping the with ultrahighenergy pulses of light that last for an extreely short aount of tie. These short pulses scrable the interior of a cell without causing it to explode, as long pulses do. We can odel a typical such cell as a disk 5.0 µ in diaeter, with the pulse lasting for 4.0 ns with a power of.0x10 1 W. We shall assue that the energy is spread uniforly over the faces of 100 cells for each pulse. a) How uch energy is given to the cell during this pulse? b) What is the intensity (in W/ ) delivered to the cell? c) What are the axiu values of the electric and agnetic fields in the pulse? Recall that power is energy/tie. So.0x10 1 W is.0x10 1 Joules/sec. Energy ( J s ) ( s) J 8000J This is the total energy, which is spread out over 100 cells, so the energy for each individual cell is 80 Joules.

20 Exaple: High-Energy Cancer Treatent Scientists are working on a technique to kill cancer cells by zapping the with ultrahighenergy pulses of light that last for an extreely short aount of tie. These short pulses scrable the interior of a cell without causing it to explode, as long pulses do. We can odel a typical such cell as a disk 5.0 µ in diaeter, with the pulse lasting for 4.0 ns with a power of.0x10 1 W. We shall assue that the energy is spread uniforly over the faces of 100 cells for each pulse. a) How uch energy is given to the cell during this pulse? b) What is the intensity (in W/ ) delivered to the cell? c) What are the axiu values of the electric and agnetic fields in the pulse? To get intensity, we need to divide power/area. The area for a cell is just the area of a circle: Area r ( )

21 Exaple: High-Energy Cancer Treatent Scientists are working on a technique to kill cancer cells by zapping the with ultrahighenergy pulses of light that last for an extreely short aount of tie. These short pulses scrable the interior of a cell without causing it to explode, as long pulses do. We can odel a typical such cell as a disk 5.0 µ in diaeter, with the pulse lasting for 4.0 ns with a power of.0x10 1 W. We shall assue that the energy is spread uniforly over the faces of 100 cells for each pulse. a) How uch energy is given to the cell during this pulse? b) What is the intensity (in W/ ) delivered to the cell? c) What are the axiu values of the electric and agnetic fields in the pulse? To get intensity, we need to divide power/area. The area for a cell is just the area of a circle: Area r ( ) Now divide to get intensity: Intensity Power 100 r W W This is the total area of all 100 cells.

22 Exaple: High-Energy Cancer Treatent Scientists are working on a technique to kill cancer cells by zapping the with ultrahighenergy pulses of light that last for an extreely short aount of tie. These short pulses scrable the interior of a cell without causing it to explode, as long pulses do. We can odel a typical such cell as a disk 5.0 µ in diaeter, with the pulse lasting for 4.0 ns with a power of.0x10 1 W. We shall assue that the energy is spread uniforly over the faces of 100 cells for each pulse. a) How uch energy is given to the cell during this pulse? b) What is the intensity (in W/ ) delivered to the cell? c) What are the axiu values of the electric and agnetic fields in the pulse? To get the field strengths, recall our intensity forula: I = 1 ε 0cE ax E ax = I ε 0 c = 10 1 ( )( ) = V B ax = E ax c = T

23 Energy and oentu in EM Waves EM waves also carry oentu. This eans 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. Radiation Pressure = S av c This is the sae as the total energy absorbed by the surface. If the energy is reflected, the pressure is doubled.

24 Exaple: Solar Sails Suppose a spacecraft with a ass of 5,000 kg has a solar sail ade of perfectly reflective aluinized fil with an area of.59x10 6. 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? Assue that at the earth s distance fro the sun, the intensity of sunlight is 1410 W/.

25 Exaple: Solar Sails Suppose a spacecraft with a ass of 5,000 kg has a solar sail ade of perfectly reflective aluinized fil with an area of.59x10 6. 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? Assue that at the earth s distance fro the sun, the intensity of sunlight is 1410 W/. Radiation Pressure = S av c = 1410 W s = Pa Recall that Pressure = Force/Area. We can use this and F=a to get our forula: P F F A a F P A a F a P A Since the sunlight reflects fro our solar sail we should double the given pressure. a ( N.5 10 ).59 4 kg s

26 Standing EM Waves When EM waves are reflected we can have a superposition of waves traveling in opposite directions, foring a STANDING WAVE. After cobining the forulas for the opposite-directed waves, and applying a bit of trigonoetry, we arrive at forulas for the E and B fields of a standing EM wave. E y x, t = E ax sin kx sin(ωt) B z x, t = B ax cos kx cos(ωt) We can find the positions where these fields go to zero (at all ties t). These are called the NODAL PLANES: For the E-field we need sin(kx)=0, which leads to the following locations: x = 0, λ 3λ, λ,, λ, Siilarly for the B-field we need cos(kx)=0, which gives: x = λ 4, 3λ 4, 5λ 4,

27 Standing EM Waves If we have 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 forulas are even the sae: λ n = L n f n = n c L (n = 1,,3,.. ) (n = 1,,3,.. ) These forulas give the wavelengths and frequencies for standing waves that will fit in a box of length L