Question

An electromagnetic rail gun can fire a projectile using a magnetic field and an electric current. Consider two conducting rails that are \(0.500 \mathrm{m}\) apart with a \(50.0-\mathrm{g}\) conducting rod connecting the two rails as in the figure with Problem \(46 .\) A magnetic field of magnitude \(0.750 \mathrm{T}\) is directed perpendicular to the plane of the rails and rod. A current of \(2.00 \mathrm{A}\) passes through the rod. (a) What direction is the force on the rod? (b) If there is no friction between the rails and the rod, how fast is the rod moving after it has traveled \(8.00 \mathrm{m}\) down the rails?

Short answer

(a) The force on the rod is to the left. (b) The rod's speed after traveling 8.00 m is 15.5 m/s.

Step-by-step solution

Understanding the problem

We have a conducting rod of mass \(50.0\, \text{g} = 0.0500\, \text{kg}\) that moves on a pair of rails which are 0.500 m apart. A magnetic field of 0.750 T is applied perpendicular to the plane of the rails. A current of 2.00 A runs through the rod. We need to determine both the direction of the force on the rod and its final speed after travelling 8.00 m, assuming no friction.

Calculate the force direction

We can use the right-hand rule to determine the direction of the force. The current flows through the rod from one rail to the other, so point your right thumb in the direction of the current. The magnetic field points perpendicular to your palm (out of the surface if using standard notation), and your fingers should point in this direction. The force is directed in the direction of your palm (perpendicular to both field and current). In this case, the force will be to the left (if the magnetic field is out of the page).

Calculate the magnetic force magnitude

The magnetic force on the rod can be calculated using the formula \( F = BIL \), where \( B = 0.750\, \text{T} \), \( I = 2.00\, \text{A} \), and \( L = 0.500\, \text{m} \). Substitute these values to find \( F \):\[ F = 0.750\, \text{T} \times 2.00\, \text{A} \times 0.500\, \text{m} = 0.750\, \text{N}. \]

Calculate rod's acceleration

Use Newton's second law to find the acceleration \(a\). The net force \(F\) acts on the mass \(m\):\[ a = \frac{F}{m} = \frac{0.750\, \text{N}}{0.0500\, \text{kg}} = 15.0\, \text{m/s}^2. \]

Determine rod's final velocity

Since the acceleration is constant, use the kinematic equation \( v^2 = u^2 + 2as \), where \( u = 0\) m/s (initial velocity), \( a = 15.0\, \text{m/s}^2\), and \( s = 8.00\, \text{m} \):\[ v^2 = 0 + 2 \times 15.0\, \text{m/s}^2 \times 8.00\, \text{m} = 240.0\, \text{m}^2/\text{s}^2. \]\[ v = \sqrt{240.0\, \text{m}^2/\text{s}^2} = 15.5\, \text{m/s}. \]

Key concepts

Magnetic Field

In the realm of physics, a magnetic field is an invisible force field that exerts a magnetic force on moving electric charges and magnetic dipoles like magnets or ferromagnetic materials. Magnetic fields are generated by electric currents or changing electric fields, with the magnetic strength denoted in Teslas (T). In our electromagnetic rail gun problem, a magnetic field of 0.750 T is applied perpendicular to the rails and the rod.
Understanding how a magnetic field works is essential because it interacts with the electric current flowing through the rod, creating the force needed to propel the projectile. When a charge moves through a magnetic field, it experiences a force which is orthogonal to both its velocity and the magnetic field direction. Thus, knowing the orientation of the magnetic field helps us predict the force direction and behavior of our system.

• The field’s perpendicular nature ensures maximum interaction, providing an ideal setup for the electromagnetic rail gun.
• A consistent magnetic field ensures a uniform force along the path, simplifying calculations.

Electric Current

An electric current is a flow of electric charge carried by moving electrons in a wire or conductor. In our problem, a current of 2.00 A flows through the conductive rod. The current is a crucial component because it interacts with the magnetic field to produce the necessary Lorentz force. The relationship between current (I), magnetic field (B), and length of conductor (L) is given by the formula: \( F = BIL \).
This interaction between the current and the magnetic field propels the rod along the rails. The direction and magnitude of the current directly affect the direction and magnitude of the force experienced by the rod.

• A higher current would result in a stronger force, accelerating the projectile more rapidly.
• The conductive paths provided by the rails ensure a stable flow of current, maintaining a constant force direction and magnitude.

Right-Hand Rule

The Right-Hand Rule is a technique used in physics to determine the direction of a force due to a magnetic field. When you place your right hand with your thumb pointing in the direction of the conventional current (positive to negative), and your fingers pointing in the direction of the magnetic field lines, your palm faces the direction of the force exerted on the conductor.
In our problem, the current flows through the rod, and the magnetic field is perpendicular to this motion (outward from the plane). By using your right hand following the rule, the force will push the rod sideways, perpendicular to both the current and field. This unique directionality allows for precise control over the projectile's motion.

• This tool helps visualize 3D interactions of current and magnetic fields, crucial for aerospace and electrical applications.
• The predictability of the force direction aids in ensuring the correct setup of our rail gun for desired motion.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause this motion. In this problem, we use kinematic equations to determine how fast the rod is moving after traveling a certain distance of 8.00 m.
Starting from rest, the rod travels under constant acceleration, which is computed using the formula \( a = \frac{F}{m} \). Knowing the acceleration, we can find the final velocity using the equation \( v^2 = u^2 + 2as \).
With this approach, we calculate the rod's final speed to be 15.5 m/s. Kinematics allows us to translate the force and acceleration into the rod's motion interpretation:

• The temporal evolution of the rod's speed gives insights into the projectile's eventual impact velocity.
• Understanding these motion equations ensures efficient and accurate predictions of projectile behavior under different conditions.