Problem 74 - University Physics with Modern Physics

Question

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 \(\vec { B }\) fills the region between the rails (Fig. \(P 27.74 )\) (a) Find the magnitude and direction of the net force on the con- ducting 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 \(B = 0.80 \mathrm { T } , \quad I = 2.0 \times 10 ^ { 3 } \mathrm { A } , \quad m = 25 \mathrm { kg }\) and \(L = 50 \mathrm { cm } .\) For simplicity assume 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

Answer

The net force is 800 N horizontally. Distance to reach escape velocity is approximately 1,960,000 m.

Step 1: Magnet Force on the Bar

To find the force, use the formula for the magnetic force on a current-carrying conductor: \( F = I \cdot L \cdot B \cdot \sin\theta \). Since the magnetic field is perpendicular to the motion, \( \theta = 90^\circ \) and \( \sin\theta = 1 \).

Step 2: Calculate the Magnetic Force Magnitude

Given the values from the problem: \( B = 0.80 \mathrm{T} \), \( I = 2.0 \times 10^3 \mathrm{A} \), and \( L = 0.50 \mathrm{m} \). The magnitude of the force is \( F = I \cdot L \cdot B = 2.0 \times 10^3 \times 0.50 \times 0.80 \). Calculating this gives \( F = 800 \mathrm{N} \).

Step 3: Direction of the Force

Using the right-hand rule, the direction of force is given by the cross product of the current and the magnetic field. In the setup, using current flow direction and magnetic field, the force will be horizontal along the rails.

Step 4: Distance for a Given Speed

The work done on the bar is converted to kinetic energy. Set the work done by the magnetic force equal to the change in kinetic energy: \( F \cdot d = \frac{1}{2}m v^2 \). Substitute \( F = 800 \mathrm{N} \) and rearrange to find \( d \): \( d = \frac{m v^2}{2F} \).

Step 5: Calculate Distance for Specific Speed

Substitute known values for the mass, initial speed, and force into the equation from Step 4. For speed \( v \): \( m = 25 \mathrm{kg} \), and using \( F = 800 \mathrm{N} \), \( d = \frac{25 \cdot v^2}{1600} \).

Step 6: Distance for Escape Velocity

Now calculate the distance required for the escape speed of \( v = 11.2 \times 10^3 \mathrm{m/s} \). Plugging into the equation: \( d = \frac{25 \cdot (11.2 \times 10^3)^2}{1600} \). Compute this value.

Key Concepts

Magnetic Force

Magnetic force is a fascinating phenomenon that is crucial in understanding how electromagnetic rail guns operate. When a current-carrying conductor such as a bar is placed in a magnetic field, it experiences a force. This is mathematically defined as \( F = I \cdot L \cdot B \cdot \sin\theta \), where:

• \( F \) is the magnetic force.
• \( I \) is the current flowing through the conductor.
• \( L \) is the length of the conductor.
• \( B \) is the magnetic field strength.
• \( \theta \) is the angle between the magnetic field and the direction of the current.

In the case of the rail gun, the magnetic field is oriented perpendicularly to the current, making \( \theta = 90^\circ \) and thus \( \sin\theta = 1 \). This simplifies the equation to \( F = I \cdot L \cdot B \).
With the given conditions, a magnetic force of \( 800 \mathrm{N} \) is exerted on the bar, propelling it along the rails. The force is crucial for the acceleration of the bar, eventually leading to high-speed motion.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When analyzing an electromagnetic rail gun, understanding how kinetic energy is developed is vital. The relationship is described by the equation for kinetic energy: \( KE = \frac{1}{2}mv^2 \).

• \( KE \) denotes the kinetic energy.
• \( m \) is the mass of the object.
• \( v \) is the velocity of the object.

As the conducting bar moves along the rail, the work done by the magnetic force is transformed into kinetic energy. This work can be defined as the force multiplied by the distance (\( F \cdot d \)), and it must equal the kinetic energy gained by the bar:
\( F \cdot d = \frac{1}{2}mv^2 \).
From this equation, we can determine how far the bar must travel to achieve a certain speed by rearranging it to calculate the distance \( d \). This demonstrates how energy conversion is utilized to accelerate objects in rail guns.

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from the gravitational pull of a celestial body, such as Earth. For Earth, the escape velocity is a high \( 11.2 \mathrm{km/s} \), demonstrating the significant energy required.
Applying this to the electromagnetic rail gun, the goal would be for the bar to reach this velocity to theoretically launch a projectile into orbit. The necessary distance for the bar to travel along the rails to achieve this speed can be determined using the kinetic energy equation:
\( d = \frac{m \times v^2}{2F} \).
In the problem, substituting the values of mass, escape velocity, and the magnetic force gives the distance needed. Calculating for this scenario provides an understanding of the enormous energy requirements for reaching such high speeds, a core challenge in adapting rail gun technology for space applications.

Right-Hand Rule

The right-hand rule is a helpful mnemonic for visualizing the direction of force in magnetic systems. When dealing with concepts like the electromagnetic rail gun, understanding this rule aids in determining the direction of the magnetic force exerted on the bar.

• To apply the rule, align your right thumb with the direction of the conventional current flow, \( I \).
• Extend your fingers in the direction of the magnetic field, \( B \).
• Your palm will point in the direction of the resultant force, \( F \).

In the setup of a rail gun, the current and magnetic field interact perpendicularly, leading to a force that acts horizontally along the length of the bar. This directionality determined by the right-hand rule is essential for ensuring the bar moves in the desired path, facilitating a deeper comprehension of electromagnetic interactions and their practical engineering applications.