Moving glass slab* A slab of glass moves to the right with speed v. A flash of light is emitted

Moving glass slab* A slab of glass moves to the right with...

Key Concepts

Special Relativity

Special relativity is the framework that governs the behavior of objects and signals (like light) when observed from different inertial frames moving at a constant relative velocity. It introduces key effects such as time dilation and length contraction, and it maintains the invariance of the speed of light in vacuum for all observers, which deeply influences how we analyze moving systems and interactions with signals traveling through different media.

Relativistic Velocity Addition

Relativistic velocity addition is a method used to combine velocities in a manner that is consistent with special relativity. Unlike classical addition of velocities, this formula ensures that the resultant speed never exceeds the speed of light, a requirement that is crucial when calculating the effective speed of light propagating in a medium that is itself moving relative to an observer.

Time Dilation and Length Contraction

Time dilation and length contraction are phenomena predicted by special relativity. Time dilation means that time intervals can differ between observers moving relative to one another, while length contraction implies that the measured length of objects in the direction of motion appears shorter in a moving frame. Both concepts are essential for analyzing scenarios where distances and time intervals need to be translated between different inertial frames.

Optical Media and Refractive Index

The refractive index of an optical medium quantifies how much the medium slows down the speed of light compared to its speed in vacuum. In such media, the speed of light is reduced to c/n, where n is the refractive index. Understanding how this reduced speed interacts with the relative motion of the medium is key to analyzing light propagation in moving materials.

Fresnel Drag Effect

The Fresnel drag effect describes how the speed of light is modified when it travels through a medium that is in motion relative to an observer. This effect accounts for the dragging of the light by the medium, and its derivation involves applying both the refractive properties of the medium and the relativistic transformations that arise from the medium's motion. It is an important concept when considering the propagation of light in moving materials.

Transcript

00:01 Our question says that a slab of glass with an index of refraction in moves to the right with speed v.

00:08 A flash of light is emitted at point a and says to look at figure 18 to get an image of this, and passes through the glass arriving at point b a distance l away.

00:17 The glass has thickness d in the reference frame where it is at rest, and the speed of light in the glass is c divided by in.

00:24 How long does it take the light to go from point a to point b, according to the observer at rest with respect to a and b? check your answers for the case of v equals c, v equals 0, and n equals 1.

00:37 Okay, so we're going to assume that the left edge of the glass is even when the flash of light is emitted.

00:45 So there is no loss of generality with that assumption.

00:48 And we do the calculation in the frame of reference in which point a and point b are at rest.

00:53 And the glass is then moving to the right with speed v.

00:56 If the glass is not moving, we would not have this motion result.

01:00 Or we would have this no motion result, where t of v is equal to zero, so we'll write t of v is equal to zero is equal to the time in the glass, time in the vacuum.

01:33 We'll call this t sub vu or va for vacuum.

01:41 Okay? and this is equal to, well, this is going to be equal to the distance in the glass divided by the speed in the glass.

01:49 So this is going to be equal to d.

01:53 We're not a room there.

01:54 So this is going to be equal to d divided by, let's write this down just a little bit.

02:01 So it actually looks like a d divided by the velocity of the speed in the glass.

02:07 So we'll call this v sub g lass.

02:13 We'll actually just call it gl for glass.

02:15 So that's the speed in the glass.

02:18 Plus, now this is going to be the distance in the vacuum divided by the speed in the vacuum.

02:25 So the distance in the vacuum is going to be l minus d divided by the speed in the vacuum, which is just c, the speed of light.

02:36 Okay.

02:37 So then this is equal to, because we were told that this is going to be d.

02:43 We were told that the velocity in the glass is just equal to c divided by n and then of course still plus l minus d over c okay so we can simplify this just a little bit more and this is going to be n times d over c minus l minus d over c okay so c is the common denominator so this is l plus n minus one times d divided by c, the speed of light.

03:31 Okay.

03:32 Well, if the index of refraction is n equals 1, then the glass will have no effect on the light, and the time would synthesize divided by the speed of light.

03:44 So we'll write for t of n equals to 1, this is equal to the time in the glass.

03:56 We're just going to abbreviate this as gl now, plus the time in the vacuum.

04:02 T v u for vacuum okay well the time in the glass is equal to the um excuse me the time of the glass is equal to distance in the glass divided by the speed in the glass and the time the distance in the vacuum divided by the speed in the vacuum well if n is equal to one we're still going to have d but the speed in the glass is now c okay and then we still have plus l minus d divided by c.

04:39 Okay.

04:40 So this is just equal to l over c, if n is equal to 1.

04:48 All right.

04:49 So now let us consider the problem from a relativistic point of view.

04:53 So the speed of light in the glass will be the relativistic sum of the speed of light in the stationary glass, c over n, and the speed of the glass, v by equation 36 -7a.

05:04 And then we're also going to redefine a term and call it beta, which you will, which you will see here.

05:10 So the speed of light in the glass, we'll call this, the speed of light, we'll call it c in glass.

05:25 So that's the speed of light in the glass, gl, is equal to c over in plus v, divided by, where v is the speed the c times v, divided by n, okay, so we'll simplify this a little bit.

05:57 This is c over n plus v divided by 1 plus v over n c.

06:11 So that's a pretty simple simplification.

06:13 That squared cancels with that c.

06:16 Okay.

06:17 So this is equal to, we're going to pull c over n out front, 1 plus vn over c, divided by 1.

06:37 Plus v over n c okay so now as i said we're going to read we're going to define a new term called beta and that beta term is just going to be what's in parentheses there so we're just going to call this c over n times beta where beta is equal to one plus vn over c divided by one plus vn over nc so c over n times beta all right so the contracted width of the glass from the earth frame of reference then is given by equation 36 -3a.

07:17 So we can write that, i'm going to run out of room on this page.

07:20 So we're going to start a new page here.

07:22 So we're going to call this the distance and the moving glass.

07:26 We'll call this moving glass gl is equal to the distance divided by gamma.

07:42 So this is d over gamma, where gamma is one over the square root of one minus v squared over c squared.

07:50 So this is the square root of 1 minus v squared over c squared times the distance d.

08:08 Okay.

08:09 So we assume that the light enters the block with the left edge of the block and write a simple equation for the displacement of the leading edge of the light and the leading edge of the block, then equal and solve for the time when the light exists or excuse me when the light exits to the right of the block.

08:28 So the distance by the light and the block will call this x light is equal to the velocity of light in glass.

08:53 Well, we've been using c for the abbreviation of light.

08:56 So let's stick with that.

08:56 So this is x of c...

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