Set A: Units and Dimensions

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Energy times time. Momentum times length. Magnetic flux times charge. And torque times time. Four completely different things, right? Now tell me: how many of them have the same dimensions as Planck's constant? One? Two? We will check at the end. And look, JEE Main loves questions like this. Today you get two ways to crack them, and the traps to avoid. So, let's begin.

0:34Way one: write everything in M, L, T, A

So, way one is the honest way. Write every quantity in mass, length, time and current, that is M, L, T and A, okay? Energy is M L squared T to the minus two. Pressure is M, L to the minus one, T to the minus two. Planck's constant, small h, is M L squared T to the minus one. The square brackets around h on screen just mean "the dimensions of h", okay? And see, only the power of T differs from energy. Keep this one in mind; it comes back at the end. Two more for electricity and magnetism. Charge is current times time, so A times T, okay? And the magnetic field B is M, T to the minus two, A to the minus one. Clear? Now let's use them. Take e B over m e: the electron's charge, times a magnetic field, divided by the electron's mass. What are its dimensions? Look. On top, A T, times M, T to the minus two, A to the minus one. The current cancels. Then divide by mass, and the M cancels too. What is left? Only T to the minus one! See how simple that was? T to the minus one is a frequency: the cyclotron angular frequency, omega equals q B over m, with q the charge and m the mass. So way one always works. But it is slow, right? One slip in a power, and the answer is gone.

2:11Way two: spot a quantity you already know

Now the topper's way. Before breaking anything into MLT, ask: do I already know a piece of this? Take epsilon nought A V over d: A is an area, d a distance, and V a potential difference. Epsilon nought A over d is a capacitance, C, isn't it? So the whole thing is C times V, a charge, and its unit is the coulomb. One line! Next, half L I squared, divided by half C V squared. Both of these are energies, okay? Energy divided by energy: dimensionless. That's it, no calculation at all! Next, magnetic moment times magnetic field, the little arrows on m and B showing they are vectors. The energy of a magnet in a field is U equals minus m dot B, okay? So m times B is simply an energy: M L squared T to the minus two. Now my favourite. Mu nought, times epsilon nought, times E over B, where E is an electric field and B a magnetic field. Most students start writing MLTA for all four. But wait: do you already know any of these pieces? Yes! Mu nought epsilon nought is one over c squared, where c is the speed of light. And E over B is c itself, okay? So the product is one over c squared, times c: just one over c. One over a speed: L to the minus one, times T, okay? That is the topper's trick: spot the known pieces first, and grind only what is left. With me?

3:50JEE traps: the look-alikes

Now, where do students lose marks? In the look-alikes. First, the pair R C and L over R: both of these are times, okay? Remember this pair. So, what about R C over L? R C is a time, L over R is a time, so R C over L is dimensionless, right? No, no, careful! This is the JEE trap. Look at the bottom: only L, not L over R, right? So write R C over L as R C divided by L over R, then times one over R, okay? Time over time cancels, leaving the dimensions of one over resistance. Not a pure number at all! Got it? Next one. Planck's constant divided by a wavelength, h over lambda. Is it a force, or a momentum? Let's see. M L squared T to the minus one, divided by L, gives MLT to the minus one: a momentum. Force is MLT to the minus two. One power of T different, so h over lambda and force do not match! Last look-alike: magnetic flux and potential difference. Both appear together in every induction question, so same dimensions, right? No, no! Magnetic flux is voltage times time: one extra factor of time. So they can never match. Is that clear?

5:12Example 1: three formulas that all give a speed

Now for Example one, an easy warm-up. Show that root of P over rho, root of Y over rho, and root of T over mu all have the dimensions of speed. P is pressure, rho is density, Y is Young's modulus, T is tension, and mu is mass per unit length. Pause the video, and try it first. Done? Okay, let's check. Pressure is M, L to the minus one, T to the minus two; density is M, L to the minus three. Divide: the M cancels, leaving L squared T to the minus two. Young's modulus has the same dimensions as pressure, so Y over rho gives the same thing, okay? Now root T over mu. Careful with this letter T, okay? Is it a time? No! Here T is tension, a force. Always check what each letter means. So tension, MLT to the minus two, divided by mass per length, M L to the minus one. The M cancels, and L over L to the minus one gives L squared. Again, L squared T to the minus two! Now take the square root of L squared T to the minus two. L T to the minus one: metres per second, a speed. All three are speeds. Nice and simple!

6:36Example 2: which quantities match Planck's constant

And now Example two: our puzzle from the start. Which of these have the dimensions of Planck's constant? Option a, energy times time. Option b, momentum times length. Option c, magnetic flux times charge. Option d, torque times time. Pause and try. Hint: Planck's constant is M L squared T to the minus one; write each option the same way. Done? Okay, let's check. Option a: energy, M L squared T to the minus two, times a time. The power of T goes from minus two to minus one: M L squared T to the minus one. It matches! Option b: momentum is MLT to the minus one. Times a length, so one more power of L, okay? M L squared T to the minus one. Matches again! Option c looks scary, right? Flux is M L squared, T to the minus two, A to the minus one, and charge is A times T, okay? The current cancels, and T to the minus two times T gives T to the minus one. And look, this matches too! Now option d, torque times time. Many students strike it off: torque is not an energy, so this cannot be Planck's constant, right? No, no, careful! Torque has the dimensions M L squared T to the minus two, exactly like energy, though it is not an energy. Times a time: M L squared T to the minus one. Matches! Dimensions never see the name of a quantity. So the answer is: all four! Remember our puzzle? Four totally different-looking things, all with the dimensions of Planck's constant. That is exactly why JEE loves these questions.

8:30Quick revision: two ways to find dimensions, and the traps

So, quick revision. Way one: write each quantity in M, L, T and A, and cancel. Way two, the topper's way: spot the known pieces first, like C V, half L I squared, mu nought epsilon nought equals one over c squared, and R C, which is a time. And watch the look-alikes: h over lambda is a momentum, and flux is voltage times time. Now go and try the practice problems on this page. You will find them easy now, I promise. All the best!