Question

A rail gun accelerates a projectile from rest by using the magnetic force on a current-carrying wire. The wire has radius \(r=5.10 \cdot 10^{-4} \mathrm{~m}\) and is made of copper having a density of \(\rho=8960 \mathrm{~kg} / \mathrm{m}^{3}\). The gun consists of rails of length \(L=1.00 \mathrm{~m}\) in a constant magnetic field of magnitude \(B=2.00 \mathrm{~T}\) 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

Question: A copper wire with a cross-sectional radius of 5.10 × 10^{-4} m is placed on horizontal rails with a length of 1.00 m. The magnetic field between the rails is 2.00 T directed vertically downward. A current of 1.00 × 10^4 A flows through the wire. The wire starts from rest at one end of the rails. Calculate the final speed of the wire as it leaves the rails. (The density of copper is 8960 kg/m³.) Answer: Step 1: Calculate the mass of the wire. mass = 8960 kg/m³ * (π * (5.10 * 10^{-4} m)^2 * 1.00 m) ≈ 0.0459 kg Step 2: Find the magnetic force on the wire. magnetic force = 2.00 T * 1.00 * 10^4 A * 1.00 m = 2.00 * 10^4 N Step 3: Calculate the acceleration of the wire. a = F / m ≈ (2.00 * 10^4 N) / (0.0459 kg) ≈ 4.36 * 10^5 m/s² Step 4: Determine the final speed of the wire. v^2 = 0 + 2 * a * L ≈ 2 * (4.36 * 10^5 m/s²) * (1.00 m) v^2 ≈ 8.73 * 10^5 m²/s² v ≈ √(8.73 * 10^5 m²/s²) ≈ 934 m/s The final speed of the wire as it leaves the rails is approximately 934 m/s.

Step-by-step solution

Calculate the mass of the wire

First, we need to find the mass of the wire. To do that, we can use the following formula: mass = density * volume To find the volume of the wire, we can use the formula for the volume of a cylinder: volume = π * r^2 * L Where, r is the radius of the wire, and L is the length of the rails. The mass of the wire can then be calculated as: mass = density * (π * r^2 * L) Using the given values, let's plug in the numbers: mass = 8960 kg/m³ * (π * (5.10 * 10^{-4} m)^2 * 1.00 m) Calculate the mass of the wire.

Find the magnetic force on the wire

The magnetic force acting on a current-carrying wire can be found using the equation: magnetic force = B * I * L Where B is the magnetic field, I is the current flowing through the wire, and L is the length of the wire. Using the given values: magnetic force = 2.00 T * 1.00 * 10^4 A * 1.00 m Calculate the magnetic force on the wire.

Calculate the acceleration of the wire

Now, we can use Newton's second law of motion to find the acceleration of the wire: F = m * a Where F is the magnetic force, m is the mass of the wire, and a is the acceleration. We can re-write the equation to solve for the acceleration (a): a = F / m Using the calculated values from Steps 1 and 2, plug in the numbers. Calculate the acceleration of the wire.

Determine the final speed of the wire

Finally, we can use the kinematic equation to find the final speed (v) of the wire: v^2 = u^2 + 2 * a * s Where u is the initial speed, a is the acceleration, and s is the distance (length of the rails). The initial speed is 0 m/s since the wire starts from rest. The formula becomes: v^2 = 0 + 2 * a * L Using the calculated values from Step 3 for acceleration (a) and the given value for the length of the rails (L = 1.00 m), plug in the numbers. Solve for the final speed (v) of the wire. Don't forget that you need to take the square root to find v.