A copper rod of mass m slides under gravity on two smooth parallel rails, with separation $l$ and set at an angle of $\theta$ with the horizontal. At the bottom, rails are joined by a resistance R. There is a uniform magnetic field B normal to the plane of the rails, as shown in the figure. The terminal speed of the copper rod is : 
To determine the terminal speed of the copper rod, we need to understand the forces and electromotive forces (emf) at play. Here’s a step-by-step analysis and derivation:
- The copper rod is sliding down due to gravity along inclined smooth rails. The force due to gravity along the incline is given by: F_{\text{gravity}} = mg \sin \theta
- As the rod slides through the magnetic field B, an emf is induced due to the motion of the rod in a magnetic field. This emf, according to Faraday’s law, is given by: \text{emf} = B \cdot v \cdot l where v is the velocity of the rod and l is the length of the rod.
- This induced emf will cause a current to flow through the resistance R at the bottom, resulting in a magnetic force that opposes the motion (Lenz's Law). The current I is given by: I = \frac{\text{emf}}{R} = \frac{B \cdot v \cdot l}{R}
- The magnetic force opposing the motion of the rod can be expressed as: F_{\text{magnetic}} = I \cdot l \cdot B = \frac{B^2 \cdot v \cdot l^2}{R}
- At terminal velocity, the gravitational force component along the incline is balanced by the magnetic force: mg \sin \theta = \frac{B^2 \cdot v \cdot l^2}{R}
- Solving for the terminal velocity v gives: v = \frac{mgR \sin \theta}{B^2 l^2}
Thus, the terminal speed of the copper rod is:
This matches the correct answer option.
Similar Questions
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Important Questions from Electricity and Magnetism
- A short bar magnet placed with its axis at $30^\circ$ with an external field of 800 Gauss, experiences a torque of 0.016 N.m. The work done in moving it from most stable to most unstable position is $\alpha \times 10^{-3} \text{ J}$. The value of $\alpha$ is ______.
- An electromagnetic wave of frequency $100\text{ MHz}$ propagates through a medium of conductivity, $\sigma = 10\text{ mho/m}$. The ratio of maximum conduction current density to maximum displacement current density is __________.
$\left[ \text{Take } \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\text{ Nm}^2\text{/C}^2 \right]$ - An inductor stores $16\text{ J}$ of magnetic field energy and dissipates $32\text{ W}$ of thermal energy due to its resistance when an a.c. current of $2\text{ A}$ (rms) and frequency $50\text{ Hz}$ flows through it. The ratio of inductive reactance to its resistance is _________. ($\pi = 3.14$)
- A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^\circ$ with vertical, then maximum induced EMF between the point of suspension and point of oscillation is ___________ mV. (Take $g = 10\text{ m/s}^2$)
Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is 50 mA for $V = 5\sin(100t)$ The values of L and R are shown in the figure. The capacitance of the capacitor (C) used is ___________ $\mu\text{F}$.
