Translating Rods and Self-Torques
G.R.Dixon, April 28, 2009
Fig. 1 depicts a rigid rod, moving in the x direction of inertial frame K at constant speed v. The rod is compressed by charges q1 and q2. There is no rotation; the torque around the rodís midpoint (for example) is zero. (The exaggerated non-rectangular cross section is a length contraction effect.)
Compressed Rod in K
Let us consider the Lorentz force on q2. The E field of a charge like q1, moving with constant velocity, is radial and has components
There is also a magnetic field at q2:
The Lorentz force on q2 accordingly has components
Now if the apparatus were at rest in K, the line of action of each electric force would lie parallel to the rod. But when the apparatus moves as indicated, the Lorentz (electric + magnetic) force acting on either charge has a line of action that does not lie parallel to the rod. The Lorentz force engenders a CCW torque about the rodís midpoint. Yet, the entire apparatus does not rotate in the xy plane.
Evidently there must be a counteracting torque. And here we encounter a profoundly interesting characteristic of moving, compressed rods: when inclined to their direction of motion, such rods engender "self-torques" that act upon whatever compresses them. (Similar comments apply to stretched rods.) The self-torques precisely cancel any "Lorentz" torques, and consequently the rods and compressing entities do not rotate as they translate.
Let us conclude by considering a compressed rod that is not inclined to its direction of motion. We recognize two cases: (1) the rod lies perpendicular to its line of motion, and (2) the rod lies parallel to its line of motion.
Fig. 2 depicts the first case. The force, that each charge exerts on its end of the rod, is less than it would be if the rod were at rest. To the extent the rod is considered to be a stiff spring, the spring constant is less by virtue of the rodís motion!
Fig. 3 depicts the second case. There is no magnetic force on either charge, and the electric force is the same as when the apparatus is at rest. But when moving, the rod is length contracted. The spring constant is accordingly greater than it is when the rod is at rest.