Dipole potential: Numerical

let's work out some numericals on electric potential due to dipoles so here's the first one we are given a dipole there is a point dipole tiny dipole whose moment is given to be 3 times 10 to the minus 8 coulomb meter this is not centimeter this is coulomb meter and we are asked to calculate what the potential is going to be a distance of 2 meters from that dipole along the axis so how do we do this well in a previous video we already derived what the expression for the potential due to a dipole anywhere is so let me quickly show you that we saw that if you have a dipole let's say there's a negative charge there's a positive charge separated by some distance we used to call 2a and if you want to know what the potential at some point p at a distance r from the center center of the dipole this then we saw that that expression is going to be k q into 2 a 2 a cos theta where theta is this angle theta divided by divided by r squared and how did we derive that well we calculate what the potential here is going to be due to this charge potential here is going to be due to this charge we added them and then we um we did some mathematics where we said okay let's assume that p is to be very very far away and if you need some clarity or you need a refresher on where how did we derive that where does it all come from the great idea to positive studio uh go it'll be a great idea to go back and watch our video on the derivation of electric potential due to dipoles but anyways let's continue what's given to us is we don't we're not given um q we're not given the distance we're given the dipole moment and we're given some vector what is that well remember that this product itself is what we call this product product of charge and the distance that itself is what we call the dipole moment which we represent as p and what is that direction say direct the type of moment is taken to be a vector and its direction is always from the negative charge to the positive charge so this this is your negative charge this has got to be your positive charge so only here dipole moment is this way okay and so you know to think about that angle the theta is the angle between the dipole moment vector and the vector that points towards p the r vector and the dipole moment vector that's our angle all right so why don't you pause the video and see with this now can you go ahead and solve this particular problem all right so let's do this so what is k k is that familiar 1 over 4 pi epsilon naught whose value is 9 times 10 to the power 9. and i don't know the charge or 2a but i know the total product that's dipole moment that's given to me as 3 times 10 to the power -8 and what is cos theta over here what is theta well theta is this angle in our case notice if i were to draw that vector from here to here that angle is zero the two are aligned so cos zero just let me write that cos zero divided by r square r is this distance two meters so two square is four and it's in meters so i don't have to do any conversion and there therefore all i have to do now is solve this so i get cos 0 is 1 so this will be 27 this will be 10 divided by 4 so that's 270 by 4 that gives me 4 6 are 24 3 carries 34 7's are 28 2.5 there you have it 67.5 volt that's the potential here we're getting a positive sign does that make sense it's positive yeah because the point is closer to the positive charge and a little farther away from the negative charge so it makes sense that we're getting a positive voltage let's try a second one again we are given dipole this time we have charges given the distance between the dipole is given we have to find the potential at point p at a distance 10 meters from the dipole and again notice that this is a point dipole this distance is much smaller compared to this so we can use the far away approximation the same thing that we did here so again can you pause the video and try solving this one on your own all right let's do this so we'll just substitute this value so you get vp is equal to k is 1 by 4 pi epsilon naught that's 9 times 10 to the power 9 q is the charge that's given then remember we just have to calculate put in the magnitude of the charge the sign is already baked in when we derive the formula so just the magnitude so 3 times 10 to the power minus 6 minus 6 because it's a micro micro is 10 to the minus 6 and times 2a now one of the mistakes i used to make while substituting over here is 2 a i used to put 2 times say 1 1 millimeter but it's not this itself is 2 here remember 2a is the distance between the two charges okay so 1 millimeter itself is my 2a so it's 1 and a milli milli is 10 to the minus 3. so let's take care of ours let's take care of our 10 powers times cos theta what is cos theta well there's a 60 so we might write cos 60 that's wrong it's not cost 60. can you pause and think a little bit about why it's not cost 60 all right let's see what is theta again remember theta is the angle between the dipole moment vector which starts from negative charge to the positive charge and the r vector so over here what is our dipole moment vector so dipole moment vector is from negative charge to the positive charge it's downwards this way this is our dipole moment vector and our r vector is this way towards p and so notice the angle is the angle theta is this angle and that's not 60 the whole angle is 180 so this is 180 minus 60. so that's our theta 180 minus 60. so let's correct that so that gives us 180 i i mean i can just write 120 but um i like to do i like to calculate this way using our l what do you call it allied angles yeah and hit angles i think that's the term but anyways so divided by r square what is r r is 10 meters so it's going to be it's going to be 10 square that's 100. so let's see we just now have to plug in so 9 times 3 is 27 times this this and this cancels out see 10 to the power 9 minus 6 minus 3 minus 9 that nicely cancels out what is cos of 180 minus 16 well i remember the other angles thing if cos of 180 minus theta is since one this is in the second quadrant i write a s t c so in second quadrant cos is negative so it's going to be minus cos 60. so this is going to be minus cos 60 and cos 60 is half so i'll get minus half over here divided by 100 and that's going to be 27 by 2 which is what is that um 17 18.6 no no 13.5 so it's negative 13.5 divided by 100 so that's minus 0.135 volt and so let's think about does it make sense that we're getting a negative sign over here yeah because this time notice that point p is a little closer to the negative charge compared to the positive charge so even if we had wrongly substituted this as cos 60 and gotten a positive value we could just look over here and say hey that's supposed to be negative and then i could have like realized why am i getting a positive value and i could have gone back and corrected myself