 Okay, now let us talk about since we are talking about magnetic field lines property Let me talk about flux magnetic flux write down magnetic flux, okay now if I take Enclosure if I take an enclosure Just like I take for Gauss law like 3d enclosure Okay, so if I take an enclosure like this How much do you think magnetic flux should be? How much it should be? First of all magnetic flux is a measure of what? Can you type in what does it measure measure of what what magnetic flux means? Okay. Tell me what? correct just like electric flux is number of Electric field lines Okay passing through passing Through an area Similarly magnetic field is proportional to number of Magnetic lines magnetic field lines That are passing through an area Okay, now If I take If I take an area in such a way that it is an enclosure Okay, if it is an enclosure Okay, and inside the enclosure you have total charge of plus q Then what will be the mag what will be the electric flux equals to? Electric flux is q by epsilon naught This is what Gauss law has told us Isn't it Now can I find out? What is the magnetic flux through an enclosure electric flux through an enclosure is? How much is a charge enclosed divided by slim not now? What is the value of magnetic flux through and 3d enclosure that is equal to what can you guess? It will be zero magnetic flux Through an enclosure is always zero Okay, why what is the reason for this? What do you think is the reason for this if you try to Understand in terms of the you know in terms of correlating the situation with the Gauss law if you try to correlate You know there is no analog of a single charge in magnetic field But then there is an analog of a dipole So if you enclose plus q and minus q that is equivalent to enclosing a bar magnet Let's say north and south pole Okay, so if you enclose both plus q and minus q which is a dipole electric flux will be zero Because q enclose will be zero similarly It happens here. You cannot enclose just north pole or south pole. You have to enclose both So if you enclose both Then the flux should be equal to zero Okay, so that is one way of understanding Okay, second way of understanding is that Suppose magnetic field lines Get generated from the north pole Okay, it get generated So one field line goes Outside the enclosure, but then one field line will come inside the south pole also Okay, so net net the number of field lines that are going away From this enclosure is zero because one goes out and one comes in So net net it is zero if you enclose a bar magnet like this Fine the strength of north pole will be exactly equal to the strength of south pole Whenever they both exist Okay, so that is a number of field line Going from the north pole will always be equal to number of field line that are coming towards the south pole That is one another way to understand suppose you do not enclose any magnet inside If you do not enclose any magnet inside then magnetic field line Since they are forming a loop will go in and will come out Okay, it will be like this So it goes in magnetic field line. It has to form a loop So that is why if it goes in it has to come out as there is no pole There is no poles inside had there been a south pole this field line would have Captured by the south pole, but then since there is no Magnet inside that is why the number whatever field line goes in Has to come out since it has to form a loop So in all cases the total flux inside a 3d enclosure will be always zero Okay, but then You remember that There is a magnetic flux we discussed in previous chapter through An area okay like in magnetic induction what we have found out emf is rate of change of magnetic flux Okay, with a negative sign over here. How come this is there? We have just proved that magnetic flux should be zero so if it is zero Then Phi B is zero so rate of change of Phi B should also be zero because it is zero. It is a constant so how come D Phi by DT is non-zero any answer see flux Through an enclosure is zero Okay, but when we find emf we find flux through What? through a loop Okay, flux through a loop is different from flux through an enclosure Enclosure is 3d enclosure it it covers entire You know volume But if you have a loop it could be just a two-dimensional thing like this Okay, so if it is like that net flux need not be zero magnetic field could you know come from You know below and move up like this Okay, so net flux is not zero now But if you have a sphere, okay, which is a 3d enclosure net flux is zero Okay, so it's not that it is a generic rule that flux magnetic flux should always be zero No magnetic flux through an enclosure is zero magnetic flux through a loop is simply B Dot da Integral and we have done it in a great detail Okay, so this was the magnetic flux and rate of change of this through a loop is emf fine So do not confuse the flux in a loop with flux through an enclosure Okay, now till now we have been talking about Magnetic field due to a bar magnet in a very qualitative manner We have not yet introduced any mathematical relation or anything that has to do with calculation of Magnetic field due to a bar magnet Okay, so Let us try to see how we can derive the equation for magnetic field due to a bar magnet Okay, so what I'm trying to do here is that I am saying bar magnet is Equivalence or equivalent to a solenoid Why I am saying so You can just visualize a Solenoid how it is You have a solenoid. Let us say a solenoid is this Okay, I can just draw a small cylinder, you know and say that consider it as a solenoid so Let's take this as a solenoid Okay, can you imagine field on this imagine it is wrapped around like this, you know it is wrapped It's a solenoid current is flowing like this number of turns per unit length is n Okay, can you try to draw the magnetic field lines? because of solenoid Just imagine we have already done this Remember that You'll see that magnetic field line inside the solenoid will be You know like this It tends to be straight Isn't it it will be like this Okay, and if it is an ideal solenoid, you know if it is an ideal solenoid then inside the solenoid Magnetic field lines will be uniform magnetic field lines per unit area Inside the solenoid will be constant and magnetic field will be a constant inside if it is a Perfect solenoid and outside the solenoid magnetic field line will be zero Okay, but when we say that magnetic field line in Outside the solenoid is zero Okay, we are assuming that the length of the solenoid goes on and on Okay, but that is not the actual case We take a fixed length of cylinder here and you'll see that Magnetic field line will take a turn like this You know and form a loop Like that all of you understanding this any doubt on this particular thing Feel free to message Go in like this So this is how magnetic field lines will be as if as if Here is the North Pole This is North Pole and as if this is a South Pole Getting it. So you'll see that magnetic field lines of a solenoid of a finite solenoid is similar to magnetic field line due to a bar magnet write down the field lines due to solenoid is similar to that of bar magnet okay so since the situation is analogous with respect to field lines and ultimately the magnetic field is Determined by how the field lines are arranged because the magnetic field is consequence of Exactly how the field line is and what is the density of field line? Okay, so that is what determines magnetic field at any point in time or at any point in the space Okay, so I can say here that they these two are analogous, okay, and What is a basic quantity in the magnetic field in the magnetic field? dipole Or to be more precise, I can write down magnetic dipole Fine, so magnetic dipole is something which connects This scenario with that scenario why I am saying so because Since field lines are similar and if I say that this solenoid has magnetic dipole moment m Okay, if this solenoid has magnetic dipole m and at a distance x At a distance x if its magnetic field is B then if I take Then if I Take a bar magnet if I take a bar magnet of same dipole moment Will it have same magnetic field at that location or not? Yes or no, right? So You know it it's like this you take You take two electrons, okay you take two electrons and magnetic field sorry the electric field due to those two electron if the Magnitude of that is E Okay, and rather than taking two electrons what you do you take helium plus two Okay, if you take helium plus two or Just to make sure that Or you take any atom with minus two charge, okay Let's not try to equate positive and negative ones take two electron and take a non-metal with minus two Charge in it They will both have same electric field because electric field depends on What is the charge? How much is a charge and? How it is distributed? so both the cases you take two electrons or a full atom with Minus two charge both are equivalent to minus two charge minus two electron charge Which is a point charge, okay? They are equivalent similarly here Okay, similarly here you have a Bar magnet and a solenoid, okay? They both will have the same magnetic field if they both have same magnetic dipole moment Fine because magnetic dipole moment is the most basic element in the magnetic field All right Right, so without further ado. Let us now try to find out the magnetic Field due to a solenoid in terms of magnetic dipole moment Right, so I am now trying to find magnetic field B as a function of magnetic dipole moment Okay, so that when I get the magnetic field as a function of Magnetic dipole moment then it doesn't matter whether there is a solenoid or a Piece of bar magnet if bar magnet has a dipole moment M I'll use the same function and put the value of magnetic dipole moment there to get the magnetic field Okay