A lightweight metal rod of length ℓ and mass m can slide without friction on two horizontal, parallel, perfectly conducting rails that are separated by the same length ℓ. The rails are connected at the far end by a resistor of resistance R. A uniform magnetic field of magnitude B is directed vertically downward throughout the region containing the rails and the rod.
At time t = 0 the rod is given an initial rightward speed v₀ and then released so that no external horizontal forces act on it afterward.
(1) Obtain an expression for the magnitude of the horizontal magnetic force acting on the rod in terms of v, B, ℓ, and R while the rod is moving with speed v.
(2) Using Newton’s second law, derive a differential equation governing the rod’s speed as a function of time, and solve this equation to find v(t).
(3) Determine the characteristic time constant τ that describes the exponential decay of the rod’s speed.
(4) Derive an expression for the total horizontal distance the rod travels from t = 0 until it effectively comes to rest, and discuss how this distance depends on the resistance R.