"Problem 4.5 In Fig. 4.6 ##p_{1}## and ##p_{2}## are (perfect) dipoles a distance r apart. What is the torque on ##p_{1}## due to ##p_{2}##? What is the torque on ##p_{2}## due to ##p_{1}##? [In each case I want the toruqe on the dipole about its own centre. If it bothers you that the answers are not equal and opposite, see Prob. 4.29.]" (Introduction to Electrodynamics, 3rd edition by David J. Griffiths; pg. 165)
Because I have not included Fig. 4.6 I will attempt to describe it (I also think it's safe to guess that most people own a copy of this text aha...).
##p_{1}## is on the left pointing upwards and ##p_{2}## is to its right pointing right; they are separated by a distance r.
Here's what I've done so far:
##\tau## = ##p## x ##E##
##E_{dip}(r)## = ##\frac{p}{4\pi \epsilon_{o} r^{3}} (2 cos\theta \hat{r} + sin\theta \hat{\theta})##
To find the toque on 2 from 1 I first find the electric field produced by 1 using the above formula.
##E_{2}## = ##\frac{p_{1}}{4 \pi \epsilon_{o} r^{3}} \hat{\theta}## (since the angle between these two vectors is ##\frac{\pi}{2}##.
Then I can find the torque by apply the above formula and voila!
The part I'm having difficulty with is that when I have to find the torque on the second dipole to the first. The question asks for the torque about its own centre...and I'm not sure what to do with that; but what I first tried was to "move" the dipoles in vector space and rotate them so that the second was pointing upwards at the origin and then did the same thing I did above. No matter how I think about it the angle ##\theta## in the formula for the electric field is ##\frac{\pi}{2}## (or some integer multiple of it).
However, the solution manual uses a value of ##\pi## for ##\theta## in calculating this second electric field and I really don't understand why. It certainly results in the torques not being equal and opposite - but I can't grasp it. Any help understanding this would be much appreciated. Thanks a bunch in advance!
Because I have not included Fig. 4.6 I will attempt to describe it (I also think it's safe to guess that most people own a copy of this text aha...).
##p_{1}## is on the left pointing upwards and ##p_{2}## is to its right pointing right; they are separated by a distance r.
Here's what I've done so far:
##\tau## = ##p## x ##E##
##E_{dip}(r)## = ##\frac{p}{4\pi \epsilon_{o} r^{3}} (2 cos\theta \hat{r} + sin\theta \hat{\theta})##
To find the toque on 2 from 1 I first find the electric field produced by 1 using the above formula.
##E_{2}## = ##\frac{p_{1}}{4 \pi \epsilon_{o} r^{3}} \hat{\theta}## (since the angle between these two vectors is ##\frac{\pi}{2}##.
Then I can find the torque by apply the above formula and voila!
The part I'm having difficulty with is that when I have to find the torque on the second dipole to the first. The question asks for the torque about its own centre...and I'm not sure what to do with that; but what I first tried was to "move" the dipoles in vector space and rotate them so that the second was pointing upwards at the origin and then did the same thing I did above. No matter how I think about it the angle ##\theta## in the formula for the electric field is ##\frac{\pi}{2}## (or some integer multiple of it).
However, the solution manual uses a value of ##\pi## for ##\theta## in calculating this second electric field and I really don't understand why. It certainly results in the torques not being equal and opposite - but I can't grasp it. Any help understanding this would be much appreciated. Thanks a bunch in advance!
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