Problem 72 - 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 \(\overrightarrow{\boldsymbol{B}}\) fills the region between the rails (Fig. 27.63\()\) . (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 \(B=0.50 \mathrm{T}, \quad I=2.0 \times 10^{3} \mathrm{A}\) \(m=25 \mathrm{kg},\) 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

(a) 500 N to the right, (b) \( \frac{v^2}{40} \) m, (c) 3136000 m.

Step 1: Identify the Magnetic Force

The magnetic force on the conducting bar due to the magnetic field is given by the Lorentz force formula: \( F = I L B \). In this formula, \( I \) is the current, \( L \) is the length of the conductor, and \( B \) is the magnetic field strength. The force is perpendicular to both the current direction and the magnetic field.

Step 2: Calculate the Force Magnitude

Substitute the given values into the formula \( F = I L B \): \( I = 2.0 \times 10^{3} \, \text{A} \), \( L = 0.50 \, \text{m} \), \( B = 0.50 \, \text{T} \). Thus, \( F = 2.0 \times 10^{3} \, \text{A} \times 0.50 \, \text{m} \times 0.50 \, \text{T} = 500 \, \text{N} \).

Step 3: Determine the Force Direction

The direction of the force is given by the right-hand rule. If you point your thumb in the direction of current and your fingers in the direction of the magnetic field, your palm will face in the direction of the force. Here, the force will be perpendicular to both the current and the magnetic field, typically to the right if the current is to the right.

Step 4: Find the Distance to Attain Speed v

Using the work-energy principle \( W = \Delta KE \), and noting that work done by the magnetic force is \( F \cdot d \), and \( \Delta KE = \frac{1}{2}mv^2 \), we set \[ Fd = \frac{1}{2}mv^2 \]. Substitute \( F = 500 \, \text{N} \) and solve for \( d \): \[ d = \frac{m v^2}{2F} \].

Step 5: Calculate d for General Speed v

Given \( m = 25 \, \text{kg} \) and the expression from Step 4, we substitute these values to calculate generic \( d \): \( d = \frac{25 \, \text{kg} \times v^2}{2 \times 500 \, \text{N}} = \frac{25 v^2}{1000} = \frac{v^2}{40} \).

Step 6: Find Distance for Escape Speed

Substitute the escape speed \( v = 11.2 \, \text{km/s} = 11200 \, \text{m/s} \) into the equation from Step 5: \( d = \frac{11200^2}{40} = \frac{125440000}{40} = 3136000 \, \text{m} \).

Key Concepts

Magnetic Force Calculation

When examining an electromagnetic rail gun, one of the primary calculations involves determining the magnetic force exerted on a conducting bar. This force is crucial as it dictates the amount of acceleration the bar experiences within the rail gun.

The magnetic force in this context is calculated using the Lorentz force formula, expressed as:

\[ F = I L B \]

Here:

• \( I \) is the current flowing through the rails, given in amperes (A).
• \( L \) is the length of the conducting bar, measured in meters (m).
• \( B \) represents the magnetic field strength, in tesla (T).

By substituting the known values, the force can be easily determined. For the provided example with \( I = 2.0 \times 10^3 \, \mathrm{A} \), \( L = 0.50 \, \mathrm{m} \), and \( B = 0.50 \, \mathrm{T} \), the force is calculated as:

\[ F = 2.0 \times 10^3 \, \text{A} \times 0.50 \, \text{m} \times 0.50 \, \text{T} = 500 \, \text{N} \]

This force acts perpendicular to both the direction of current and the magnetic field. Understanding this relationship is fundamental in utilizing magnetic force to propel objects in a rail gun.

Lorentz Force

The Lorentz force is a key concept in electromagnetism and a fundamental aspect of understanding the operation of a rail gun. This force arises from the interaction between the electric current and the magnetic field, producing motion in the conducting object.

The direction of the Lorentz force can be found using the right-hand rule. In this rule, if you imagine pointing your thumb in the direction of the current flow and your fingers in the direction of the magnetic field, the force will appear to emanate from the palm. It acts perpendicular to both the current's direction and the magnetic field.

For example, in a rail gun, when the current flows to the right and the magnetic field points upwards, the resulting force will push the conducting bar in a perpendicular direction, typically outwards along the rails.

This perpendicular interaction is what makes the Lorentz force unique and instrumental for technologies like rail guns, where rapid acceleration is needed through electromagnetic means. By comprehensively understanding the Lorentz force, one can better grasp how rail guns can be designed to propel objects at enormous speeds.

Escape Velocity

Escape velocity refers to the minimum speed that an object needs to "escape" from the gravitational influence of a planet or celestial body without further propulsion. On Earth, this speed is about 11.2 kilometers per second (km/s).

In the context of electromagnetic rail guns, achieving escape velocity means propelling a payload with sufficient speed to leave Earth's gravitational pull. This is a challenging task, but theoretically possible by using rail guns that apply a constant electromagnetic force over a distance.

The necessary distance to achieve escape speed for a fixed force can be calculated by rearranging the work-energy principle equation, setting the work done by the force equal to the kinetic energy required for escape:

\[ d = \frac{m \cdot v^2}{2F} \]

Here:

• \( m \) is the mass of the object.
• \( v \) is the escape velocity.
• \( F \) is the force exerted by the rail gun.

Substituting the escape velocity \( v = 11200 \, \text{m/s} \) and the magnetic force \( F = 500 \, \text{N} \), we find that the distance \( d \) required is approximately 3,136,000 meters. This gives a sense of the immense power and lengths needed to accelerate objects to escape speed using only electromagnetic forces.

Work-Energy Principle

The work-energy principle is a fundamental concept that connects the work done by forces on an object to its change in kinetic energy. This principle is especially useful in solving problems involving electromagnetism and rail guns.

The work-energy principle is defined as:

\[ W = \Delta KE \] where \( W \) is the work done, and \( \Delta KE \) is the change in kinetic energy.

In the case of the electromagnetic rail gun, the work done by the magnetic force on the bar is \( F \cdot d \), and the change in kinetic energy is the difference from rest to its final velocity, normally calculated by \( \frac{1}{2}mv^2 \). Therefore, the equation becomes:

\[ Fd = \frac{1}{2}mv^2 \]

Utilizing this equation, one can find the distance necessary for the bar to accelerate to a given speed. By substituting known values for mass, desired speed, and force, it allows for the straightforward calculation of the distance.

This principle provides an intuitive connection between the exerted force, the distance traveled, and the final velocity achieved in the system. Hence, it's an essential tool for calculating the dynamics of systems like electromagnetic rail guns.