Electric and Magnetic Quantities

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take a resistance R and multiply it by a capacitance C, two purely electrical quantities, okay? What kind of quantity comes out? Something electrical, surely, right? No! It comes out as a time, like the reading on your stopwatch. Wait, how can that be? By the end of this class, you will prove it in a few seconds. And look, JEE loves asking which combination has the dimensions of time. So, let's begin.

0:34Two anchors: charge and potential

So, in mechanics we needed only mass, length and time, written M, L and T, okay? Now electricity enters, so we add one more base quantity, current. Its dimension is capital A, for ampere. And these square brackets mean the dimension of. Clear? Anchor number one: charge. Charge is current into time, q equals I t. This double arrow means gives us. So charge is simply A T, current into time. Really simple, right? Anchor number two: potential. V equals W by q, work divided by charge. Work, like all energy, is M L squared T to the minus two, and we divide it by A T, okay? So, the power of T in the answer? Minus two, right? No, no, careful! This is the trap. Charge also carries a T, right? Dividing by that T takes the power from minus two down to minus three. So potential is M L squared T to the minus three, A to the minus one. And emf, being a potential, has the same dimension. These two are our anchors; everything else today is built from them. Got it?

1:40Chain the rest from the anchors

Alright, now we chain the rest. Resistance is V by I, potential divided by current. So divide the potential by one more A, and the power of A goes from minus one to minus two. So resistance is M L squared T to the minus three, A to the minus two. That's it, done. Capacitance: C equals q by V, charge over potential. Dividing by the potential flips all its powers: M to the minus one, L to the minus two, T to the three, A to the one. Then multiply by A T from the charge, so T becomes four and A becomes two. With me? Electric field. The force on a charge is q into E, so E is force divided by charge. MLT to the minus two, divided by A T, gives MLT to the minus three, A to the minus one. See? That T to the minus three again. Now inductance, from emf equals L into d I by d t, the rate of change of current. But wait. Is this L the length L of our dimensions? No! Here L is inductance. Same letter, different meaning, okay? So inductance is emf into time, divided by current: M L squared T to the minus two, A to the minus two. Nice and straightforward.

3:01Magnetic quantities

Now, magnetism. The key is the force on a moving charge: F equals q v B, okay? So B is force divided by q into v. At the bottom, A T into L T to the minus one; the T cancels, leaving just A L, okay? Force divided by A L: M T to the minus two, A to the minus one. Even the L cancels! Nice, right? Magnetic flux, phi, this Greek letter, is B into area. Area is L squared, so flux is M L squared T to the minus two, A to the minus one. Next, magnetic moment: m equals I A, okay? Current has dimension A, and there is another A in the formula, so the answer is A squared, right? Oh no, that's the trap! Here this A is area, not ampere. Area is L squared, so magnetic moment is L squared A, current into area. Same in flux. In JEE, this one letter confuses many students. Not you, okay?

4:02Topper's trick: energy formulae

Now a topper's trick. Look at these three: half C V squared, half L I squared, and q squared over two C, okay? What do they have in common? Think, think. Yes, exactly! All three are energy, M L squared T to the minus two. So each one is a shortcut: know the energy, and you get C or L in one line. And the half and the two? Pure numbers, no dimensions. Just ignore them. So, let's use it right now.

4:32Example 1: capacitance and inductance from energy

Example one, an easy one. Find the dimensions of capacitance from U equals half C V squared, and of inductance from U equals half L I squared. Here U is energy, okay? Pause the video, and try it first. Done? Okay, let's check. Drop the half, so C equals U by V squared. Energy on top; at the bottom, the potential, whole squared. Now, the squaring. Quick, what do you get? Look, many students double only the L and the T, and leave the M alone. Don't! Every power doubles. So the bottom is M squared, L to the four, T to the minus six, A to the minus two. Divide: M to the minus one, L to the minus two, T to the four, A squared. The same answer as from q by V! Two roads, one answer. That is how you check yourself in the exam. Inductance is quicker: energy divided by I squared, that is, by A squared. M L squared T to the minus two, A to the minus two, matching the emf route. Absolutely perfect.

5:38Example 2: RC, L/R and root LC as time, E/B as velocity

Now a harder one, our puzzle from the start. Show that R C, L by R and root L C have the dimensions of time, and that one over root mu naught epsilon naught, and E by B, are velocities. Mu naught is the permeability and epsilon naught the permittivity, these Greek letters with a small zero. Pause the video and try it. Hint: use the defining equations. Alright, R C first. Should we multiply the two long formulae for R and C? We could, but that is slow, and one slip ruins it. Smarter: write R as V by I, and C as q by V, okay? This dot means multiply. Now the V cancels, and R C equals q by I, okay? And q by I is what? Time, yes! Because q equals I t, anchor number one. Puzzle solved! L by R: M L squared cancels, A to the minus two cancels, and T to the minus two over T to the minus three leaves T to the one. Time again! Root L C? Just a small trick: L C equals L by R, into R C, okay? Time into time is T squared, and its square root is a time. Three out of three! Now velocity. c squared equals one over mu naught epsilon naught, where c is the speed of light. So one over root mu naught epsilon naught is just c, a velocity. Simple as that. Last, E by B, okay? The force on a charge is q E from the electric field, and q v B from the magnetic field. Put them equal; the q cancels, and E by B equals v, a velocity. See how fast that was?

7:21Quick revision: two anchors and the energy shortcut

So, quick revision. Two anchors: charge is A T, and potential is M L squared T to the minus three, A to the minus one. Chain the rest: R is V by I, C is q by V, and B comes from F equals q v B, okay? And the shortcut: half C V squared, half L I squared and q squared over two C are all energy. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!