CP CALC Terminal Speed. A bar of length L=0.36 mis free to slide without friction on horizontal

CP CALC Terminal Speed. A bar of length L=0.36 mis free to...

CP CALC Terminal Speed. A bar of length $L=0.36 \mathrm{mis}$ free to slide without friction on horizontal rails, as shown in Fig. $\mathrm{P} 29.60 .$ There is a uniform magnetic field $B=1.5 \mathrm{T}$ directed into the plane of the figure. At one end of the rails there is a battery with emf $\mathcal{E}=12 \mathrm{V}$ and a switch. The bar has mass 0.90 $\mathrm{kg}$ and resistance 5.0$\Omega$ , and all other resistance in the circuit can be ignored. The switch is closed at time $t=0 .$ (a) Sketch the speed of the bar as a function of time. (b) Just after the switch is closed, what is the acceleration of the bar? (c) What is the acceleration of the bar when its speed is 2.0 $\mathrm{m} / \mathrm{s} ?$ (d) What is the terminal speed of the bar?

CP CALC Terminal Speed. A bar of length $L=0.36 \mathrm{mis}$ free to slide without friction on horizontal rails, as shown in Fig. $\mathrm{P} 29.60 .$ There is a uniform magnetic field $B=1.5 \mathrm{T}$ directed into the plane of the figure. At one end of the rails there is a battery with emf
$\mathcal{E}=12 \mathrm{V}$ and a switch. The bar has mass 0.90 $\mathrm{kg}$ and resistance
5.0$\Omega$ , and all other resistance in the circuit can be ignored. The switch is closed at time
$t=0 .$ (a) Sketch the speed of the bar as a function of time. (b) Just after the switch is closed, what is the acceleration of the bar? (c) What is the acceleration of the bar when its speed is 2.0 $\mathrm{m} / \mathrm{s} ?$ (d) What is the terminal speed of the bar?

Key Concepts

Ohm’s Law in Circuit Dynamics

Ohm’s Law, which relates voltage, current, and resistance, is fundamental in analyzing the electrical circuit involved in this problem. The battery provides a constant EMF, while the moving conductor develops an induced EMF that opposes the battery’s voltage. The net voltage divided by the resistance determines the current in the circuit, and this current, in turn, interacts with the magnetic field to produce the Lorentz force.

Terminal Velocity in Electromagnetic Systems

Terminal velocity in this context is achieved when the magnetic force that opposes the motion of the bar, generated by the induced EMF, exactly balances the driving force due to the battery’s EMF. At terminal speed, the net force on the bar is zero, so its acceleration also becomes zero. This equilibrium condition is a result of the feedback mechanism between the increasing velocity, the growing induced EMF, and the corresponding magnetic force acting against further acceleration.

Motional Electromotive Force (EMF)

This concept explains that when a conductor moves through a magnetic field, an EMF is induced along its length. The magnitude of the motional EMF is determined by the product of the magnetic field strength, the velocity of the conductor, and the effective length of the conductor within the field. It is a direct consequence of Faraday’s law of induction and represents the voltage generated due to the motion of the conductor relative to the magnetic field.

Lorentz Force

The Lorentz force describes the force experienced by a current-carrying conductor in a magnetic field. The force is given by the product of the current, the magnetic field, and the conductor’s length, and it acts perpendicular to both the current flow and the magnetic field direction. This force is essential in determining the acceleration of the conductor and plays a key role in electromagnetic damping and motion control in such systems.

Transcript

00:01 So in this problem, we have a loop that's built from this metal bar on two railings.

00:13 So there's a battery supplying 12 volts.

00:17 There's a switch.

00:19 And this bar has a mass of 0 .90 kilograms.

00:25 And it supplies the resistance of 5 oms.

00:28 It has a length of 0 .36 meters.

00:32 And through the entire area, there's a magnetic field of 1 .5 tesla's.

00:41 So first, what we want to do is determine the graph of the speed as a function of time.

00:54 So what's the speed of this bar as a function of time? first question to think about is, why would it even move at all? so when the switch closes, when this switch closes, there's going to be a current.

01:12 So suppose the switch closes, there's going to be a current like this.

01:18 So that current is going to flow clockwise.

01:23 But if you have a current carrying wire in a magnetic field, it feels a force.

01:31 And that force is i, l, b, when the current and magnetic field, are perpendicular.

01:38 And you can see that they are because the current is straight down and the magnetic field is straight out.

01:45 What's the formula for this current? well, let's call this the battery current.

01:49 So the battery current comes from ohms law.

01:54 So it's just the battery voltage divided by r.

02:00 So that current leads to this kind of force, which pushes the bar this way.

02:11 But once the bar starts moving, the area subtended by the circuit, this area right here, increases, right? because the bar is going this way, so the area is getting bigger and bigger.

02:29 But if the area is getting bigger, then the flux is changing.

02:33 And if the flux is changing, there's an induced current.

02:35 So there's another current coming from faraday's law.

02:40 So first of all, we have the induced emf is the number of turns times the rate of change of flux.

02:48 There's only one turn, so n is one, and the magnetic field is constant.

02:54 So this is just going to be b times d a d t.

02:58 Right, so i just use the fact that flux is equal to magnetic field times area.

03:05 What's the area? well, the area of the circuit is the length times x where, let's say, x is this distance.

03:21 So we weren't actually given that in the problem, but it doesn't matter.

03:25 Because the length of the bar isn't changing.

03:28 So this is just bl, dx, dt, and dx, dt is exactly what we want.

03:35 It's the speed.