 Fine, so there is one part of our section in this chapter which talks about force on a current Right, we have learned that a moving charge will experience a force Right, how much is that movie charge experiences a force of magnitude? qv cross b Right now when when I say that current is there, okay Does it mean that charges are moving? Yes or no? Which charge is moving? Positive or negative? Negative charge is moving and positive charge is stationary Alright, so charge is moving So if charge experiences a force Then automatically the wire which has the current will experience a force Isn't it? Now it is difficult to measure the charge That is why one of the reasons charge is not the fundamental unit The current is the fundamental unit Right, current is very easy to measure Okay, emitter or galamometer whatever it is You can measure the current very accurately So for the practical purposes it is easy for us to deal with current And to find out how much force is there on a current So we would ideally like to know the force on a wire that contains a current i Fine So we will try to use this expression itself to derive the force on a straight wire Which has a current i Suppose there is a wire like this Current is i Cross section area is a Number of free charges Current volume is n The length of the wire is l Okay And inside the drift velocity is vd Okay, drift velocity is like that Current will be in opposite direction Because it is electron Can you get this expression of force In terms of all these things vd, a, l and n Okay What else is given is Entire thing is in a magnetic field b That is perpendicular to the rod Going into the plane In terms of a, vd, small n and b and l Get the value of force The hint is The total amount of force experienced by the moving charges Will be the total force on the wire What will be q? q will be what? Here Total charge that is moving inside this wire How much is that? Small n is number of free electrons per unit volume So how much charge is there inside this wire? Number of electrons will be what? n a into l Right So total charge will be what? Number of electron into charge of one electron Getting it Now what will be their velocities? What are their velocities? Not just vd They will have a random velocity As will have a drift velocity Because the random velocity net force gets cancelled off So we do not need to consider the random velocity Only systematic velocity will have a net force Fine So we will ignore the random component We will say that this entire thing is moving with vd So charge q is moving with vd Right And this vd is perpendicular to b So vd cross b is just vd into b Fine So the force becomes what? q which is n a l e Then vd cross with b Like that Which direction the force will be? Tell me vd cross b Which direction? vd is like this Cross b It is downwards So the force is This is electron This downward force would have been on the positive charge On electron it will be opposite Fine This is the force Getting it vd cross b Down on a positive charge It is a negative charge So it will be opposite of that Fine These are electrons Fine Now if I Sir can we use the left hand rule also? Yes sir This is n e a vd l cross b Right Now what is this? Can you recall? I This is i Right But once you represent it as i You cannot have a vector inside i So what you do? You drop the direction from v Down to length You make length as a vector And what is the direction of this length vector? In the direction of current Or opposite direction of the electron velocity Getting it So if I write this as i l cross b I will get the Direction of force Fine The direction of force is in the direction of Length cross product with 93 And length vector is in the direction of current Just check whether you get the upward force Only i cross Sorry l cross b l is like this In the direction of current And b Upwards Fine In order to find the cross product Align your right hand towards the first vector And then fold your hand towards the second vector Thumb is the direction of cross product You cannot do like this Then you have to fold this way You have to do like this and like that So this expression Is what? Force on a conductor Of length l Which is in the magnitude b Till now we have been using this What if the conductor is like this How are you fine? And magnitude is going in How will you consider this? While you analyze this What you will do? The current is moving like that This is the current Is length vector constant? No Every time length vector changes Here it is in this direction There it is in that direction Here it is in this direction So what you will do? You will take small, small lengths Like dL, dL lengths If you consider small length You will treat it as A uniform direction One direction thing So you have found out An expression for a straight wire For a curved wire You need to integrate Every dL you have to see Which direction the force is And accordingly you integrate Can you doubt on this? This wire On to which current is Magnetic field is inside This is basically a semicircular wire This distance is of course Two times the radius This is the center Magnetic field is given as B Find out The total force on this Curved wire Do this first You have Current like this The length is L only And current is I Magnetic field is B Fine From here to here A to B is L And B to C is also L This is 60 degree First do this then we will go there Not getting it? Can you see that Just combination of two Straight wires Find out force independently First on this and then on that Which direction the force will be Up or down? L cross B L is down And then you move Towards the B It is like this If you do like this L Then you have to go towards B L cross then B So it will be down L cross B This is length Keep it very simple L cross B L cross B So this is the force On A B Force on B C will be Like this So total force how will you find Will this force be Equal to this force in magnitude They are equal How much is the magnitude? I L into B Because current Length and magnitude They are perpendicular to each other And you will see that This component will get cancelled off From that component Vertical component get cancelled off But this component will get added up How much is this Angle 30 degrees Draw a line like this This will be 30 degrees This is also 30 degrees I L B Cos of 30 degrees Two times of that One from here Fine So this is the total force on the wire Getting it Any wire has mass M Then this force is equal to mass M acceleration Now try to do this Consider DLDL elements Here DLM You can come You can assume it to be straight This direction of force will be on this DL On just on this DL This is the DL vector This direction It will be inverse Right, directed towards the center So can you see that This vertical component get cancelled off From another similar element Over there Getting it Whose force is also like this This component will get cancelled off Right This component that is left is horizontal one Now let us zoom this This is the situation This is DL vector Fine And You have a force That is perpendicular to DL Like this You have to take its components Along horizontal direction Like this This will be equal to 90 meters theta Theta Again So this force value is how much Let us say this is df Sorry this will be df This will be df cos theta Right So only df cos theta We have to add What did you find So Because the vertical component df sin theta Will get cancelled off Fine So df cos theta Is the force Along x axis Fine And df cos theta is what I DL Into B Into cos theta Fine And now look at here Because I B into DL cos theta What is DL cos theta So DL cos theta becomes dy This is DL This length is DL cos theta Fine So now just imagine Here there is a DL You have to consider this length This length here Every time you are adding up the vertical lens So if you keep on adding vertical lens You will become 2R When you integrate Keep on adding vertical lens The total force will become I B Into total vertical lens Which is 2R Getting it In that case also In that case also How much is this length First tell me what is this length The half of the length This is 30 degree This is L So this is what L cos 30 I B into 12 cos 30 Vertical lens So you take any shape Of the wire You take this also If you know this distance Which is L The force, net force on this will be I B into L Now At times force The magnetic field will be given Like it will be given that the magnetic field B is 2 Tesla 1 Tesla and whichever direction it is But many a times Magnetic field will not be given directly Have you calculated magnetic field Due to various shape and size Straight wire, circle, solenoid Toroid, we found magnetic field So that magnetic field Which is coming I L cross B Or Q V into B May not be directly given And it need not be You may have to find out Using those formulae But that magnetic field Should not be Because of it's own current That magnetic field Should be external magnetic field Magnetic field of other Current Can create a force On the current Magnetic field due to the second wire Or second current can Create a force on the first current Like that But magnetic field due to the first current Can have a force on the first current itself It's like you applying force on yourself They will be equal in opposite force In the third law and get nullified Become zero So next topic is just that I mean just a small example is taken As in two Infinite wires are taken like this Draw these wires They are placed parallel to each other Distance Between The wires G The current over here is I1 And current over there is I2 You need to find The force On The L length of this wire L is the length Both are infinite though See the hint is You have to find magnetic field On this wire And then use I cross B How much is the magnetic field due to I1 On to I2 Due to an infinite wire what is the magnetic field You guys practice at home Like where you just throw your books away And then one week after you come here Class without looking at What is that Magnetic field is Mu naught I1 by 2 pi d What is the direction of this magnetic field On to I2 What is the direction of this magnetic field I am asking you something You have to speak up Tell me which direction is the magnetic field On I2 Obviously what I am asking you How good What do you think I what Why you are using this I have already told you Use that curve This thing Unnecessary you are confusing yourself I don't understand What is the point of teaching so much If you are still you are using What is confusing you And what is the point of you coming here How many times I told you Put your thumb along the wire And then curve your hand Which is going inside Magnetic field on to I2 Is inside Goes in And what is the direction of force on I2 Of this magnetic field Because of this magnetic field Direction of force What is the value of force How will you find the value of force Magnetic field is this The force is No I L cross B Cross B What is the direction of force On to I2 Because of this magnetic field Left or right Just check L is along the current L cross B So these two currents Are attracting each other I1 is Putting a force this way And I2 will be like Pulling it that way So there will be equivalent So The force will be what I2 L into B And we are perpendicular to each other L is like this B into the world So this is just multiplication of length And magnetic field This is the magnitude of force So magnitude of force is I1 I2 By 2 pi D Into L Now if you increase the length This length if you increase The force will increase I am saying this length is infinite The force will also grow infinite That has no meaning So we will just talk about force per unit length So F by L Is mu naught I1 I2 divided by 2 pi D Okay This is the force per unit length Newton per meter Newton magnetic type of moment Galonometer Till you guys do practice at home All of this is useless This is you just If you do not look at it After going on This will just evaporate You are coming here is useless If you do not do any problem practice I am just introducing you a concept Real learning happens When you sit alone in your home Practicing problems