(II) (a) Show that the following combination of the three fund...

Question

(II) (a) Show that the following combination of the three fundamental constants of nature that we used in Example 1–10 (that is G, c, and h) forms a quantity with the dimensions of time:

tₚ = √Gh/c⁵

This quantity, tₚ, is called the Planck time and is thought to be the earliest time, after the creation of the Universe, at which the currently known laws of physics can be applied.

Accepted Answer

Hey, everyone in this problem, we're asked to find the dimensions of the quantity P formed by the following combination. So P is going to be the square root of one divided by epsilon knot. Mu knot. OK. Where epsilon knot is the vacuum permittivity and mu knot is the vacuum permeability. OK. Fundamental constants there, we're given four answer choices. Option A L divided by T to the exponent two thirds. Option BL divided by T squared, option C, the square root of L divided by T and option DL divided by T. So we wanna do dimensional analysis and remember that the dimensions on the left hand side have to equal the dimensions. On the right hand side is we need the dimensions of P to be the same as the dimensions on the right hand side. So let's think about the units of epsilon knot and new knot that's gonna help us find their dimensions and then we can work from there. OK. So when we have epsilon knot, uh the typical units we use are P squared per Newton meter squared. All right, let's try to break this down as much as we can into dimensions. Into fundamental units. And now a coon recall that that can be written as an amp multiplied by a second. So we have amp multiplied by second squared in the denominator, we have a Newton and recall that newtons can be written as kilogram meter per second squared. And we still have that meter squared in the denominator as well. OK. So this is a little bit messy, but let's go ahead and simplify in the numerator. We have a squared multiplied by S squared. And then we have this additional S squared that was in the denominator of the fraction in the denominator. And that's all divided by kilogram meter. All right. So if we want to look at the dimensions of epsilon knot, OK. While we have the dimension of AMP squared, we have seconds to the exponent four. OK. Seconds are a measure of time. So we have time to the exponent four divided by kilogram, which is a unit of mass and meters cubed. So a unit of length cubed, OK. So the dimensions of epsilon knot are a squared T to the exponent four divided by ML cubed. Now we're gonna do the same for Muno. OK. And with Muno, the typical units we see RT Tesla multiplied by meter divided by AMP. Recall that a Tesla can be written as a kilogram per second scored. And, and we have kilogram divided by second squared amp multiplied by meters divided by hm. Uh We're gonna simplify again. So we just have kilogram meter divided by second squared amp squared. And again, these are the typical units, but we wanna write it the dimensions. So the dimensions of Muno, we have kilogram which is a unit of mass multiplied by meters, a unit of length divided by second squared, the unit times squared multiplied by amps squared. So we're just gonna say a squared there. All right. Now, we have the dimensions of epsilon knot and mu knot. And we can look at the dimensions of P and because we know the dimensions of P must be equal to the dimensions of everything on the right hand side. OK. So we have the square root of one divided by the dimensions of epsilon, not A squared T to the exponent color divided by M oh Cute. And that's multiplied by the dimensions of Muno M oh divided by T squared A squared. All right. So it looks a little bit messy right now. We just have some fractions and we're gonna simplify and we can see that this AMP squared, the units of AMP squared or the dimensions are gonna divide up, OK. The mass terms are going to divide up. We're left with T to the exponent four divided by T squared. Well, that's gonna give us T squared and then we have L divided by L cubed which is gonna give us one over L squared. So when we write out the simplified version we have that the dimensions of P are going to be this square root of one divided by T squared, divided by L squared. We have a fraction in the denominators. We're dividing by a fraction, we can flip that fraction and multiply and get that the dimensions of P are therefore equal to the square root um L squared divided by T squared. And finally taking the square root, we get that the dimensions of P are going to be the dimensions of length divided by time. And so P is kind of like a measure of speed or velocity. All right. And that's what we were looking for. We were looking for those dimensions of P. We needed to figure out the dimensions of epsilon knot and mu knot first. And then we knew the dimensions of the left hand side of our equation had to be equal to the dimensions of the right hand side. If we take a look at our answer choices, we can see that D is the correct answer. We have the dimensions of length divided by time. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Fundamental Constants

Dimensional Analysis

Planck Time

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