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(II) In deriving Eq. $2, \quad v=\sqrt{F_{\mathrm{T}} / \mu,}$ for the speed of a transverse wave on a string, it was assumed that the wave's amplitude $A$ is much less than its wavelength $\lambda$ .Assuming a sinusoidal wave shape $D=A \sin (k x-\omega t),$ show via the partial derivative $v^{\prime}=\partial D / \partial t \quad$ that the assumption $A \ll \lambda$ implies that the maximum transverse speed $v_{\text { max }}^{\prime}$ the string itself is much less than the wavevelocity. If $A=\lambda / 100$ determine the ratio $v_{\max }^{\prime} / v.$$v=\sqrt{\frac{F_{\mathrm{T}}}{\mu}} \quad \quad \left[ \begin{array}{l}{\text { transverse }} \\ {\text { wave on a cord }}\end{array}\right]$ (2)

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$v \prime_{\max } < < v_{\text {wave}}$$\frac{\pi}{50} \approx 0.063$

Physics 101 Mechanics

Physics 102 Electricity and Magnetism

Chapter 15

Wave Motion

Periodic Motion

Mechanical Waves

Electromagnetic Waves

Rutgers, The State University of New Jersey

University of Washington

Hope College

University of Sheffield

Lectures

03:40

In physics, electromagnetic radiation (EM radiation or EMR) refers to the waves (or their quanta, photons) of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays. Electromagnetic waves of different frequency are called by different names since they have different sources and effects on matter. In order of increasing frequency and decreasing wavelength these are: radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays.

10:59

In physics, Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. They underpin all electric, optical and radio such electromagnetic technologies as power generation, electric motors, wireless communication, cameras, televisions, computers, and radar. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of these fields. The equations have two major variants. The microscopic Maxwell equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale. The macroscopic Maxwell equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale details. The equations were published by Maxwell in his 1864 paper "A Dynamical Theory of the Electromagnetic Field". In the original paper Maxwell fully derived them from the Lorentz force law (without using the Lorentz transformation) and also from the conservation of energy and momentum.

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It'S that for this equation. For this way, the assumption that the amplitude is much smaller than the place implies that the maximum velocity of the wave, which is defined as the prime equals the derivative of f d with respect to t since velocity is much smaller. Since the velocity of as a wave- so that's the transverse speed of the string is much smaller than the way velocity. So we want to show that this will imply this. So how do we do that? I think the easiest thing is to first figure out what the prime is takete. So, with respect to the ras this coefficient and negative omega is there a sine will become cosine. So it's this fate and the maximum happens when the cosine equals tective 1 is becomes omega times an so this is t prime x. So what is v v equals f times? It will be a beis 2 pi times. I sorry. This is times f times 2 times. So if any is a lot smaller zeptf times 8, because f is constate will be a lot smaller than f times lymptaso, because it's so much smaller, so adding a 2 pi actually would have hurt because 2 pi is brought 6 fit, is so much smaller. The lenten time by 6, it will still be so much smaller, so the tea is the prim dux. So that's how it much smaller than lambda indicates. The prime max is some much smaller than in other words, the transfers. Velocity of this train itself is so much smaller than the wave velocity part b says if we have a equals lambda divided by 100 point. So what will be the prime of x over for that? You could just plug into expression. So this is 2 pi. F. 8 and b is f times fenda, so this gives us 2 pi times 8. Over lepta overlap is over 100 point, so this is pi over 50, which is 0.063.

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