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An Electromagnetic Rail Gun. A conducting bar with mass \(m\) and length \(L\) slides over horizontal rails that are connected to a voltage source. The voltage source maintains a constant current \(I\) in the rails and bar, and a constant, uniform, vertical magnetic field \(\overrightarrow{\boldsymbol{B}}\) fills the region between the rails (Fig. P7.41). (a) Find the magnitude and direction of the net force on the conducting bar. Ignore friction, air resistance, and electrical resistance. (b) If the bar has mass \(m\), find the distance \(d\) that the bar must move along the rails from rest to attain speed \(v .\) (c) It has been suggested that rail guns based on this principle could accelerate payloads into earth orbit or beyond. Find the distance the bar must travel along the rails if it is to reach the escape speed for the earth \((11.2 \mathrm{~km} / \mathrm{s}) .\) Let \(B=0.80 \mathrm{~T}, I=2.0 \times 10^{3} \mathrm{~A}, m=25 \mathrm{~kg}\), and \(L=50 \mathrm{~cm}\). For simplicity asssume the net force on the object is equal to the magnetic force, as in parts (a) and (b), even though gravity plays an important role in an actual launch in space.

Step by step solution

01

Calculate Magnetic Force

The magnetic force on the bar can be determined using the formula for the magnetic force on a current-carrying conductor: \[ F = I L B \]Substitute the given values, where \( I = 2000 \) A, \( L = 0.5 \) m (converted from cm), and \( B = 0.8 \) T:\[ F = 2000 \times 0.5 \times 0.8 = 800 \text{ N} \] The direction of the force can be determined using the right-hand rule. Since the magnetic field is vertical and the current moves along the rail, the force acts horizontally along the direction of motion.

02

Determine Distance for Set Speed

To find the distance \(d\) required for the bar to reach a certain speed \(v\), we use the work-energy principle. The work done by the magnetic force is equal to the change in kinetic energy:\[ Fd = \frac{1}{2}mv^2 \]Solving for \(d\), we have:\[ d = \frac{mv^2}{2F} \]

03

Plug Values for Part (b)

The magnetic force \(F\) is already calculated as 800 N. For part (b), use the given mass \(m = 25\) kg and the required speed \(v\):\[ d = \frac{25v^2}{2 \times 800} = \frac{25v^2}{1600} \]Insert the desired speed incrementally as needed to calculate \(d\) for different speeds.

04

Calculate Distance for Escape Speed

For part (c), use the escape speed \(v = 11.2 \times 10^3\) m/s and the formula from the previous step:\[ d = \frac{25 \times (11,200)^2}{1600} \]Calculate the distance:\[ d = \frac{25 \times 125,440,000}{1600} = 1,965,000 \text{ meters} \]This is the distance the bar must travel to reach escape speed for Earth.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Force

When discussing the electromagnetic rail gun, a crucial concept is the magnetic force. This force is generated when a current-carrying conductor, such as our conducting bar, interacts with a magnetic field. The magnitude of the magnetic force can be calculated using the formula \( F = I L B \), where:

• \( F \) is the magnetic force,
• \( I \) is the current through the conductor,
• \( L \) is the length of the conductor,
• \( B \) is the magnetic field strength.

For our problem, the bar slides along the rails under the influence of this magnetic force, with the force directed perpendicular to both the magnetic field and the current. It is this force that propels the bar forward along the rails. Understanding the direction of the force in relation to the magnetic field and the current is vital, often determined using the right-hand rule.

Work-Energy Principle

The work-energy principle is fundamental in understanding how the rail gun accelerates the bar. According to this principle, the work done by the magnetic force is equal to the change in kinetic energy of the bar. Mathematically, this is expressed as \( Fd = \frac{1}{2}mv^2 \), where:

• \( F \) is the magnetic force,
• \( d \) is the distance the bar travels,
• \( m \) is the mass of the bar,
• \( v \) is the final velocity of the bar.

This equation helps to determine how far the bar needs to travel along the rail from a standstill to reach any specific speed \( v \). By rearranging the formula, we can solve for the distance \( d \) needed. This concept highlights the relationship between the work done on the bar and its resultant motion.

Escape Velocity

Escape velocity is the speed that an object must reach to break free from Earth's gravitational pull without further propulsion. For our rail gun example, this is applicable in hypothetical scenarios where the rail gun could launch payloads into space. The escape velocity for Earth is approximately 11.2 km/s. By using the work-energy principle, we can calculate how far the conducting bar must travel to achieve this velocity. Given the formula \( d = \frac{mv^2}{2F} \), and substituting \( v = 11,200 \) m/s, the required distance can be found. It's crucial to note that in real-world applications, other forces like gravity would also play a significant role in such launches.

Current-Carrying Conductor

The concept of a current-carrying conductor is central to the functioning of an electromagnetic rail gun. In our problem, the conducting bar acts as this current-carrying conductor, which slides over the horizontal rails. As electric current \( I \) flows through this bar, it interacts with the magnetic field \( B \) present between the rails.This interaction is what generates the magnetic force that propels the bar. The smooth flow of current is essential for maintaining a consistent force. Also, any resistance in the bar or the rails is ignored for simplicity in this exercise, though it might be relevant in practical scenarios.

Right-Hand Rule

The right-hand rule is a simple tool to determine the direction of the magnetic force on a current-carrying conductor. To use it, extend your right hand with your thumb perpendicular to your fingers.

• Point your fingers in the direction of the magnetic field \( \overrightarrow{B} \),
• Then, point your thumb in the direction of the current \( I \).

The force exerted on the conductor will then be directed outwards from your palm. For our rail gun example, since the magnetic field is vertical and the current moves along the rail, the magnetic force acts horizontally. This tool is invaluable for visualizing how electric and magnetic field interactions lead to motion.