 Let's solve a question on torque on dipoles in external electric fields. Here we have the torque on a dipole, which is making a 150 degree angle with a uniform electric field of magnitude 4 Newton meters. So the magnitude of the torque is 4 Newton meters and the dipole we can see in the diagram that's making an angle of 150 degrees with the electric field. And this torque is in the positive z direction. We can see that over here. This is 4k cap Newton meters. Now consider the following instant during the dipoles motion. So here we have that instant in the dipoles rotational motion. What is the magnitude of the torque on the dipole at this instant? All right. And the second part is choose the unit vector in the direction of the torque. So we need to choose one answer out of these six options. Okay, as always, hit pause, try this one on your own. Try to figure out the magnitude of the torque and also the direction in which the torque is acting at this instant. All right. Hopefully you tried this. Now, now we can begin, we can begin solving this question by recalling what was torque to begin with. We know that torque on a dipole with dipole moment, let's call that P in an external electric field. It's given by P cross, P cross E. It's P cross E. And this cross product is really P e sin theta. So this is a torque on the dipole. Now here P, the dipole moment is really constant and also the electric field is also constant. So this product P e is not really changing. P e is constant. P e is constant because we aren't changing the charges and the distance between them, which means we are not changing the dipole moment and we are not changing the electric field as well. So P e is constant and and that means that torque is really is proportional to just sin theta in this case. So for this case, we can say that a torque that is being experienced by the dipole, that's tau 1 and here the torque that is being experienced by this dipole is tau 2 and it's directly proportional to the angle that the dipole is making with the electric field. So if we divide these two, if we write tau 1 by tau 2, this is equal to sin theta 1 divided by sin theta 2 and sin theta 1 is sin 150 degrees. This is sin 150 degrees divided by sin 90 degrees because now the angle is 90. At this instant the angle is 90. This is equal to 4, tau 1 is 4, 4 divided by tau 2. So now if we if we figure out tau 2 from this, the torque that the dipole is experiencing here, that will be 4 into sin 90 divided by sin 150 degrees and and I encourage you to work this out. You will see that the torque at this instant, that's really 8 Newton meters. It's 8 Newton meters because sin 90 is 1 and sin 150 degrees is 0.5. So 4 divided by 0.5, that's 8. Okay, now let's try to figure out the unit vector that is the direction of the torque. So to begin with, torque is acting in the positive z direction. Let's try to see how. So we have this positive charge and electric field is to the right. This positive charge will experience a force in this direction and this negative charge will experience a force in the direction opposite to the electric field. So you can see these two forces, they will turn the dipole in this direction, in an anticlockwise direction. And we can try and figure out the direction of the torque by using the right hand rule. That is you can curl the fingers in an anticlockwise direction and you will see that the term points upwards, that is the positive z direction. Now in this case, let's try to figure out the direction here. Here the charge, the plus q charge, that will experience a force in this direction because the electric field is to the right and minus q will experience a force in this direction. So the dipole will turn in a clockwise manner like this. Now if we try to figure out the direction of the torque, if we use the right hand thumb rule, we can curl our fingers from P to E. And whereas P in this direction, P is always from negative to positive. So this right here is, this is the dipole moment. So here if we try to use the right hand thumb rule, we can see that we can try and curl our fingers from P to E. And when we do that, we can assume that P is going away from us into the plane of the screen and E is to the right. So when we curl our fingers like that, we will see that the thumb is pointing down which is in the negative z direction. Initially the thumb was pointing up when we curled our fingers from P to E and now the thumb is pointing down. So this is, this is minus k cap. The unit vector in the direction of the torque is minus k cap because the thumb points down which means the direction of the torque is in the negative z direction and that's minus k cap. Alright, you can try more questions from this exercise in the lesson and if you're watching on YouTube, do check out the exercise link which is added in the description.