Question

A rail gun accelerates a projectile from rest by using the magnetic force on a current-carrying wire. The wire has radius \(r=5.1 \cdot 10^{-4} \mathrm{~m}\) and is made of copper having a density of \(\rho=8960 \mathrm{~kg} / \mathrm{m}^{3}\). The gun consists of rails of length \(L=1.0 \mathrm{~m}\) in a constant magnetic field of magnitude \(B=2.0 \mathrm{~T}\) oriented perpendicular to the plane defined by the rails. The wire forms an electrical connection across the rails at one end of the rails. When triggered, a current of \(1.00 \cdot 10^{4}\) A flows through the wire, which accelerates the wire along the rails. Calculate the final speed of the wire as it leaves the rails. (Neglect friction.)

Short answer

Answer: To find the final velocity of the wire when it leaves the rails, follow these steps: 1. Calculate the mass of the wire using its dimensions and the density of copper. 2. Determine the magnetic force on the wire using the given current and magnetic field strength. 3. Find the acceleration of the wire using the magnetic force and its mass. 4. Calculate the final velocity of the wire using the acceleration and distance it traveled along the rails. By following these steps, you can determine the final velocity of the copper wire as it leaves the rails in the rail gun.

Step-by-step solution

Calculate the mass of the wire

First, we need to find the volume and mass of the wire. We know the radius (r) and length (L) of the wire, so we can find the volume (V) using the formula for the volume of a cylinder: \(V = \pi r^2 L\) Then, we can calculate the mass (m) using the density (ρ) of copper: \(m = \rho \cdot V\)

Calculate the magnetic force on the wire

Next, we need to determine the magnetic force (F) acting on the wire due to the current (I) and the magnetic field (B). The magnetic force on a current-carrying wire can be found using the formula: \(F = I(LB)\)

Calculate the wire's acceleration

Now that we have the magnetic force acting on the wire, we can use Newton's second law of motion (F=ma) to find the acceleration (a) of the wire: \(a = \frac{F}{m}\)

Calculate the final velocity of the wire

Finally, we can find the final velocity (v) of the wire using the kinematic equation, knowing the initial velocity (u=0), the acceleration (a), and the distance (L): \(v^2 = u^2 + 2aL\) Since the initial velocity (u) is 0, we only need to compute the square root of the right-hand side to find v: \(v = \sqrt{2aL}\) Once all calculations are done, we'll have determined the final velocity of the wire as it leaves the rails.

Key concepts

Current-Carrying Wire

A current-carrying wire is integral to devices like rail guns as it interacts with magnetic fields to produce motion. A wire that carries an electric current experiences a force when placed in a magnetic field due to the interaction of the magnetic field with the current. This is calculated using the formula:

• Magnetic Force, \( F = I \, (L \times B) \)
• \( I \) stands for current, \( L \) for length, and \( B \) for magnetic field strength

This phenomenon is crucial for numerous applications, including electromagnets and motors. In our problem, the current-carrying wire is part of a rail gun setup, where a significant current flows through the wire, inducing a magnetic force which propels the wire.

Copper Density

The density of a material like copper is a critical factor when calculating the mass of a wire. Density is defined as mass per unit volume and is denoted by \( \rho \). For copper, this is typically \( \rho = 8960 \, \text{kg/m}^3 \).

• Density helps determine the mass of objects using the formula: \( m = \rho \cdot V \)
• Volume \( (V) \) can be calculated for a cylindrical wire as: \( V = \pi r^2 L \)

Knowing the density, together with the geometric dimensions of the object, allows us to compute the mass, which is essential for further calculations involving forces and accelerations.

Newton's Second Law

Newton's Second Law of Motion forms the cornerstone for calculating acceleration when a force is applied. It states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass. It's represented as:

• \( F = ma \) where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration

In the context of the rail gun, once the magnetic force exerted on the wire is calculated, its acceleration can be found by rearranging to \( a = \frac{F}{m} \). This principle allows us to understand how the magnetic force accelerates the projectile along the rails.

Kinematic Equation

The kinematic equations are used to describe the motion of objects and are particularly important for finding velocities in uniform acceleration contexts. One of the primary equations is:

• \( v^2 = u^2 + 2aL \)
• Here, \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is acceleration, and \( L \) is distance

In the problem at hand, the initial velocity \( (u) \) of the wire is zero because it begins from rest. The equation simplifies to \( v = \sqrt{2aL} \), allowing us to solve for the final speed of the wire as it exits the rails using the previously calculated acceleration.