Question

A 30 g horizontal metal bar, 13 cm long, is free to slide up and down between two tall, vertical metal rods that are 13 cm apart. A 5.5×10⁻² T magnetic field is directed perpendicular to the plane of the rods. The bar is raised to near the top of the rods, and a 1.2 Ω resistor is connected across the two rods at the top. Then the bar is dropped.

What is the terminal speed at which the bar falls? Assume the bar remains horizontal and in contact with the rods at all times.

Answer 1

The terminal velocity of the bar can be calculated using Faraday's law of electromagnetic induction and by balancing the gravitational force with the Lorentz force. The induced current and Lorentz force oppose the acceleration due to gravity, leading to terminal velocity when the net force becomes zero.

Step-by-step explanation:

The student's question involves calculating the terminal velocity of a horizontal metal bar falling between two vertical metal rods within a magnetic field. Faraday's law of electromagnetic induction is vital here, which states that a changing magnetic flux through a circuit induces an emf (electromotive force) in the circuit. The induced emf leads to an induced current when there is a complete path for the current to flow. In this scenario, as the bar falls, it cuts through magnetic field lines, hence the magnetic flux through the loop formed by the bar, rods, and resistor changes. This induces an emf (E) given by E = B * l * v (where B is the magnetic field strength, l is the length of the bar, and v is its velocity). The induced current (I) can be determined using Ohm's Law, I = E / R, where R is the resistance of the circuit. The Lorentz force acting on the bar, which is a result of its motion through the magnetic field, opposes its acceleration due to gravity and eventually reaches a point where the net force is zero, leading to terminal velocity.

When analyzing similar problems across different conditions, the magnitude and direction of the induced current can vary based on the direction of the magnetic field and the motion of the conductive element. Additionally, power is dissipated as heat due to the resistance in the circuit, known as Joule heating, which can be calculated using the formula P = I^2 * R. The point at which this power equals the power provided by the force of gravity (equal to the gravitational force times the velocity) marks the terminal velocity of the conductive element. In the case of the falling bar, its terminal velocity can be calculated once the forces are balanced.