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Rail guns have been suggested for launching projectiles into space without chemical rockets. A tabletop model rail gun (Fig. P30.62) consists of two long, parallel, horizontal rails $\ell=3.50 \mathrm{cm}$ apart, bridged by a bar of mass $m=3.00 \mathrm{g}$ that is free to slide without friction. The rails and bar have low electric resistance, and the current is limited to a constant $I=24.0 \mathrm{A}$ by a power supply that is far to the left of the figure, so it has no magnetic effect on the bar. Figure P30.62 shows the bar at rest at the mid-point of the rails at the moment the current is established. We wish to find the speed with which the bar leaves the rails after being released from the midpoint of the rails. (a) Find the magnitude of the magnetic field at a distance of 1.75 cm from a single long wire carrying a current of 2.40 A. (b) For purposes of evaluating the magnetic field, model the rails as infinitely long. Using the result of part (a), find the magnitude and direction of the magnetic field at the midpoint of the bar. (c) Argue that this value of the field will be the same at all positions of the bar to the right of the midpoint of the rails. At other points along the bar, the field is in the same direction as at the midpoint, but is larger in magnitude. Assume the average effective magnetic field along the bar is five times larger than the field at the midpoint. With this assumption, find (d) the magnitude and (e) the direction of the force on the bar. (f) Is the bar properly modeled as a particle under constant acceleration? (g) Find the velocity of the bar after it has traveled a distance $d=130 \mathrm{cm}$ to the end of the rails.

   Rail guns have been suggested for launching projectiles into space without chemical rockets. A tabletop model rail gun (Fig. P30.62) consists of two long, parallel, horizontal rails $\ell=3.50 \mathrm{cm}$ apart, bridged by a bar of mass $m=3.00 \mathrm{g}$ that is free to slide without friction. The rails and bar have low electric resistance, and the current is limited to a constant $I=24.0 \mathrm{A}$ by a power supply that is far to the left of the figure, so it has no magnetic effect on the bar. Figure P30.62 shows the bar at rest at the mid-point of the rails at the moment the current is established. We wish to find the speed with which the bar leaves the rails after being released from the midpoint of the rails. (a) Find the magnitude of the magnetic field at a distance of 1.75 cm from a single long wire carrying a current of 2.40 A. (b) For purposes of evaluating the magnetic field, model the rails as infinitely long. Using the result of part (a), find the magnitude and direction of the magnetic field at the midpoint of the bar. (c) Argue that this value of the field will be the same at all positions of the bar to the right of the midpoint of the rails. At other points along the bar, the field is in the same direction as at the midpoint, but is larger in magnitude. Assume the average effective magnetic field along the bar is five times larger than the field at the midpoint. With this assumption, find (d) the magnitude and (e) the direction of the force on the bar. (f) Is the bar properly modeled as a particle under constant acceleration? (g) Find the velocity of the bar after it has traveled a distance $d=130 \mathrm{cm}$ to the end of the rails.

Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway,… 8th Edition

Chapter 30, Problem 62

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06/09/2020

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75 cm from a single long wire carrying a current of 24.0 A. The magnetic field B around a long straight wire carrying a current I at a distance r from the wire is given by Ampere's law as: \[B = \frac{{\mu_0 I}}{{2\pi r}}\] where \(\mu_0 = 4\pi \times 10^{-7} \, T Show more…

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Rail guns have been suggested for launching projectiles into space without chemical rockets. A tabletop model rail gun (Fig. P30.62) consists of two long, parallel, horizontal rails $\ell=3.50 \mathrm{cm}$ apart, bridged by a bar of mass $m=3.00 \mathrm{g}$ that is free to slide without friction. The rails and bar have low electric resistance, and the current is limited to a constant $I=24.0 \mathrm{A}$ by a power supply that is far to the left of the figure, so it has no magnetic effect on the bar. Figure P30.62 shows the bar at rest at the mid-point of the rails at the moment the current is established. We wish to find the speed with which the bar leaves the rails after being released from the midpoint of the rails. (a) Find the magnitude of the magnetic field at a distance of 1.75 cm from a single long wire carrying a current of 2.40 A. (b) For purposes of evaluating the magnetic field, model the rails as infinitely long. Using the result of part (a), find the magnitude and direction of the magnetic field at the midpoint of the bar. (c) Argue that this value of the field will be the same at all positions of the bar to the right of the midpoint of the rails. At other points along the bar, the field is in the same direction as at the midpoint, but is larger in magnitude. Assume the average effective magnetic field along the bar is five times larger than the field at the midpoint. With this assumption, find (d) the magnitude and (e) the direction of the force on the bar. (f) Is the bar properly modeled as a particle under constant acceleration? (g) Find the velocity of the bar after it has traveled a distance $d=130 \mathrm{cm}$ to the end of the rails.

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Key Concepts

Superposition Principle in Magnetism

The superposition principle states that the total magnetic field created by multiple sources is the vector sum of the magnetic fields due to each individual source. This means that when dealing with several current-carrying wires, as in the case of multiple rails, the magnetic field at any point can be calculated by summing the contributions of the fields from each wire. This principle is crucial in simplifying complex problems in electromagnetism by breaking them down into simpler parts.

Magnetic Field of a Long Wire

This concept involves determining the magnetic field generated by a current flowing through a long, straight conductor. Typically, Ampère’s law is used to derive the expression B = ??I/(2?r), where B is the magnetic field at a distance r from the wire, I is the current, and ?? is the permeability of free space. This formula is fundamental in electromagnetism because it allows for the calculation of the field strength created by currents in various configurations.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor or the direction of the force on a current in a magnetic field. In the context of electromagnetism, if you point the thumb of your right hand in the direction of the current, your fingers will curl in the direction of the magnetic field lines. This rule is essential for visualizing and working with magnetic fields in any problem involving moving charges or currents.

Lorentz Force on a Conductor

The Lorentz force law describes the force exerted on a current-carrying conductor when placed in a magnetic field. Mathematically, it is expressed as F = I(L × B), where I is the current, L is the vector length of the conductor within the field, and B is the magnetic field. This concept is fundamental in understanding how magnetic fields can do work on electrical currents, leading to applications such as rail guns and electric motors.

Kinematics under Constant Acceleration

This concept involves applying the equations of motion to an object experiencing constant acceleration, which are derived from Newton’s laws of motion. These equations relate displacement, velocity, time, and acceleration, and are used to predict the motion of objects under uniform forces. They are critical when analyzing the movement of bodies, especially in systems where the acceleration is constant due to consistent applied forces, such as those from a constant magnetic force.

Modeling Objects as Particles

This idea pertains to approximating the behavior of extended objects as if they were point masses, especially when analyzing their motion under external forces. In many problems, if the dimensions of the object are small compared to the scale of the physical system or if rotational effects are negligible, the object can be modeled as a particle. This simplifies the analysis by allowing the use of basic kinematic and dynamic equations without considering the complexities of distributed mass or rotational inertia.

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Transcript

00:01 In this pretty long problem, we have a rail gun that consists of two infinite long bars with a third bar sliding on top of them horizontally, as you can see in the figure.

00:10 And part a is asking us to find the magnitude of the magnetic field at a distance 1 .75 centimeters from a single long wire, which carries a current of 240 ampers.

00:20 There's not that much for us to do here.

00:22 We're just going to plug into our equation for the magnetic field from an infinitely long wire.

00:28 And in this case we're told that we have a current of 24 amps and a distance between our two wires of 0 .175 meters.

00:42 So when we plug all our values in, we're going to get a result of 2 .74 times 10 to the negative for tesla.

00:49 Part b says, for purposes of evaluating the magnetic field, model the rails is infinitely long, and then use the result using the result of part a.

00:57 We want to find the magnitude and direction of the magnetic field at the midpoint of the bar.

01:02 Now, so because the current is moving through the bar, only half of each rail is going to carry current in this case, since we're looking at the center point.

01:10 So, the field produced by the rail is half of what an infinitely long wire produces, and our answer is just going to be exactly half of our result from our first part.

01:21 And using the right -hand rule, we can see that this must be pointing in the negative j direction.

01:27 Now, part c asks us to argue that the value of the field is going to be the same at all positions of the bar to the right of the midpoint of the rails.

01:36 And at any other points along the bar, the field is in the same direction of the midpoint, but that is larger in magnitude.

01:43 So if we want to show this, then we are going to use the fact that the conductor produces the field that we just found in part b.

01:54 So under the assumption that the rails are infinitely long, the length of the rail to the left of the bar is not going to depend on location of the bar at all because they go on forever.

02:08 Now, part d then asks us to find the magnitude of the force on the bar...

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