Explanation 1) Concept: We have to take the derivative of equation 33 - 11 with respect to x and the derivative of equation 33 - 17 with respect to t . By comparing these two equations, we can prove that the electric field satisfies the wave equation. We can use the same concept to prove that the magnetic field satisfies the wave equation. 2) Formula: i) ∂ E ∂ x = - ∂ B ∂ t ii) c = 1 μ 0 ε 0 iii) - ∂ B ∂ x = μ 0 ε 0 ∂ E ∂ t 3) Given: E ( x , t ) and B ( x , t ) are the electric and magnetic field components. θ 4) Calculations: From equation 33 - 11 , we have ∂ E ∂ x = - ∂ B ∂ t Differentiating this with respect to x , we get ∂ ∂ x ∂ E ∂ x = ∂ ∂ x - ∂ B ∂ t ∂ 2 E ∂ x 2 = - ∂ 2 B ∂ x ∂ t … … … … … … … … … … … … … … … … … … … … … ( 1 ) Now, from equation 33 - 17 , we have - ∂ B ∂ x = μ 0 ε 0 ∂ E ∂ t Differentiating this equation with respect to t , we get ∂ ∂ t - ∂ B ∂ x = μ 0 ε 0 ∂ ∂ t ∂ E ∂ t - ∂ 2 B ∂ t ∂ x = μ 0 ε 0 ∂ 2 E ∂ t 2 … … … … … … … … … … … … … … … … … … ( 2 ) Comparing ( 1 ) and ( 2 ) , we get μ 0 ε 0 ∂ 2 E ∂ t 2 = ∂ 2 E ∂ x 2 ∂ 2 E ∂ t 2 = 1 μ 0 ε 0 ∂ 2 E ∂ x 2 But, c = 1 μ 0 ε 0 , so ∂ 2 E ∂ t 2 = c 2 ∂ 2 E ∂ x 2 Thus, the electric field satisfies the wave equation

11th Edition

ISBN: 9781119306856

Author: Halliday

Publisher: WILEY

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Question

Chapter 33, Problem 108P

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To show:

The electric and magnetic field components of a plane traveling electromagnetic wave must satisfy the wave equations

∂2E∂t2=c2∂2E∂x2 and ∂2B∂t2=c2∂2B∂x2

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Chapter 33, Problem 107P

Chapter 33, Problem 109P

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Chapter 33 Solutions

Fundamentals Of Physics - Volume 1 Only

Ch. 33 - Prob. 1Q Ch. 33 - Prob. 2Q Ch. 33 - a Figure 33-27 shows light reaching a polarizing...Ch. 33 - Prob. 4Q Ch. 33 - In the arrangement of Fig. 33-l5a, start with...Ch. 33 - Prob. 6Q Ch. 33 - Figure 33-30 shows fays of monochromatic Light...Ch. 33 - Figure 33-31 shows the multiple reflections of a...Ch. 33 - Figure 33-32 shows four long horizontal layers AD...Ch. 33 - The leftmost block in Fig. 33-33 depicts total...

Ch. 33 - Prob. 11Q Ch. 33 - Prob. 12Q Ch. 33 - Prob. 1P Ch. 33 - Prob. 2P Ch. 33 - Prob. 3P Ch. 33 - About how far apart must you hold your hands for...Ch. 33 - SSM What inductance must be connected to a 17 pF...Ch. 33 - Prob. 6P Ch. 33 - Prob. 7P Ch. 33 - Prob. 8P Ch. 33 - Prob. 9P Ch. 33 - Prob. 10P Ch. 33 - Prob. 11P Ch. 33 - Prob. 12P Ch. 33 - Sunlight just outside Earths atmosphere has an...Ch. 33 - Prob. 14P Ch. 33 - An airplane flying at a distance of 10 km from a...Ch. 33 - Prob. 16P Ch. 33 - Prob. 17P Ch. 33 - Prob. 18P Ch. 33 - Prob. 19P Ch. 33 - Radiation from the Sun reaching Earth just outside...Ch. 33 - ILW What is the radiation pressure 1.5 m away from...Ch. 33 - Prob. 22P Ch. 33 - Someone plans to float a small, totally absorbing...Ch. 33 - Prob. 24P Ch. 33 - Prob. 25P Ch. 33 - Prob. 26P Ch. 33 - Prob. 27P Ch. 33 - The average intensity of the solar radiation that...Ch. 33 - SSM A small spaceship with a mass of only 1.5 103...Ch. 33 - A small laser emits light at power 5.00 mW and...Ch. 33 - Prob. 31P Ch. 33 - Prob. 32P Ch. 33 - Prob. 33P Ch. 33 - Prob. 34P Ch. 33 - Prob. 35P Ch. 33 - At a beach the light is generally partially...Ch. 33 - Prob. 37P Ch. 33 - Prob. 38P Ch. 33 - Prob. 39P Ch. 33 - Prob. 40P Ch. 33 - A beam of polarized light is sent into a system of...Ch. 33 - Prob. 42P Ch. 33 - A beam of partially polarized light can be...Ch. 33 - Prob. 44P Ch. 33 - When the rectangular metal tank in Fig. 33-46 is...Ch. 33 - In Fig. 33-47a, a light ray in an underlying...Ch. 33 - Light in vacuum is incident on the surface of a...Ch. 33 - In Fig. 33-48a, a light ray in water is incident...Ch. 33 - Figure 33-49 shows light reflecting from two...Ch. 33 - In Fig. 33-50a, a beam of light in material 1 is...Ch. 33 - GO In Fig. 33-51, light is incident at angle 1 =...Ch. 33 - In Fig. 33-52a, a beam of light in material 1 is...Ch. 33 - SSM WWW ILW in Fig. 33-53, a ray is incident on...Ch. 33 - Prob. 54P Ch. 33 - Prob. 55P Ch. 33 - Rainbows from square drops. Suppose that, on some...Ch. 33 - A point source of light is 80.0 cm below the...Ch. 33 - The index of refraction of benzene is 1.8. What is...Ch. 33 - SSM ILW In Fig. 33-57, a ray of light is...Ch. 33 - In Fig. 33-58, light from ray A refracts from...Ch. 33 - GO In Fig. 33-59, light initially in material 1...Ch. 33 - GO A catfish is 2.00 m below the surface of a...Ch. 33 - In Fig. 33-60, light enters a 90 triangular prism...Ch. 33 - Suppose the prism of Fig. 33-53 has apex angle =...Ch. 33 - GO Figure 33-61 depicts a simplistic optical...Ch. 33 - Prob. 66P Ch. 33 - GO In the ray diagram of Fig. 33-63, where the...Ch. 33 - a At what angle of incidence will the light...Ch. 33 - Prob. 69P Ch. 33 - In Fig. 33-64, a light ray in air is incident on a...Ch. 33 - Prob. 71P Ch. 33 - An electromagnetic wave with frequency 4.00 1014...Ch. 33 - Prob. 73P Ch. 33 - A particle in the solar system is under the...Ch. 33 - SSM In Fig, 33-65, a light ray enters a glass slab...Ch. 33 - Prob. 76P Ch. 33 - Rainbow. Figure 33-67 shows a light ray entering...Ch. 33 - The primary rainbow described in Problem 77 is the...Ch. 33 - SSM emerges from the opposite face parallel to its...Ch. 33 - Prob. 80P Ch. 33 - Prob. 81P Ch. 33 - Prob. 82P Ch. 33 - SSM A ray of white light traveling through fused...Ch. 33 - Three polarizing sheets are stacked. The first and...Ch. 33 - In a region of space where gravirational forces...Ch. 33 - An unpolarized beam of light is sent into a stack...Ch. 33 - SSM During a test, a NATO surveillance radar...Ch. 33 - The magnetic component of an electromagnetic wave...Ch. 33 - Calculate the a upper and b lower limit of the...Ch. 33 - In Fig. 33-71, two light rays pass from air...Ch. 33 - Prob. 91P Ch. 33 - In about A D 150, Claudius Ptolemy gave the...Ch. 33 - Prob. 93P Ch. 33 - Prob. 94P Ch. 33 - Prob. 95P Ch. 33 - Prob. 96P Ch. 33 - Two polarizing sheets, one directly above the...Ch. 33 - Prob. 98P Ch. 33 - Prob. 99P Ch. 33 - Prob. 100P Ch. 33 - Prob. 101P Ch. 33 - Prob. 102P Ch. 33 - Prob. 103P Ch. 33 - Prob. 104P Ch. 33 - Prob. 105P Ch. 33 - In Fig. 33-78, where n1 = l.70, n2 = l .50, and n3...Ch. 33 - When red light in vacuum is incident at the...Ch. 33 - Prob. 108P Ch. 33 - SSM a Show that Eqs. 33-1 land 33-2 satisfy the...Ch. 33 - Prob. 110P

Knowledge Booster

The electric and magnetic field components of a plane traveling electromagnetic wave must satisfy the wave equations ∂ 2 E ∂ t 2 = c 2 ∂ 2 E ∂ x 2 and ∂ 2 B ∂ t 2 = c 2 ∂ 2 B ∂ x 2

Solution Summary: The author explains that the electric and magnetic field components of a plane traveling electromagnetic wave must satisfy the wave equations.

To show:

The electric and magnetic field components of a plane traveling electromagnetic wave must satisfy the wave equations

∂2E∂t2=c2∂2E∂x2 and ∂2B∂t2=c2∂2B∂x2

Want to see the full answer?

Chapter 33 Solutions

The electric and magnetic field components of a plane traveling electromagnetic wave must satisfy the wave equations ∂ 2 E ∂ t 2 = c 2 ∂ 2 E ∂ x 2 and ∂ 2 B ∂ t 2 = c 2 ∂ 2 B ∂ x 2

Solution Summary: The author explains that the electric and magnetic field components of a plane traveling electromagnetic wave must satisfy the wave equations.

To show: The electric and magnetic field components of a plane traveling electromagnetic wave must satisfy the wave equations ∂2E∂t2=c2∂2E∂x2 and ∂2B∂t2=c2∂2B∂x2

Want to see the full answer?

Chapter 33 Solutions

To show:

The electric and magnetic field components of a plane traveling electromagnetic wave must satisfy the wave equations

∂2E∂t2=c2∂2E∂x2 and ∂2B∂t2=c2∂2B∂x2