Question

A conducting rod of length \(L\) slides freely down an inclined plane, as shown in the figure. The plane is inclined at an angle \(\theta\) from the horizontal. A uniform magnetic field of strength \(B\) acts in the positive \(y\) -direction. Determine the magnitude and the direction of the current that would have to be passed through the rod to hold it in position on the inclined plane.

Short answer

Answer: To find the magnitude of the current needed to hold the rod in position, we can use the formula I = (m * g * sin(theta)) / (L * B), where m is the mass of the rod, g is the acceleration due to gravity (9.81 m/s^2), theta is the angle of inclination of the plane, L is the length of the rod, and B is the magnetic field strength. To ensure the magnetic force opposes the gravitational force, the direction of the current should be such that it produces a force in the x-direction opposite and upslope to the gravitational force while keeping the rod in position.

Step-by-step solution

Analyze the forces acting on the rod

We have two forces acting on the rod: gravitational force (Fg) acting downward in the x-z plane, and magnetic force (Fm) acting parallel to the x-axis due to the current through the rod in the presence of the magnetic field. The rod will be held in position if these two forces balance out, i.e., if Fm = Fg.

Express the gravitational force

The gravitational force on the rod (Fg) acting parallel to the x-z plane can be found using the following equation: Fg = m * g * sin(theta) Here, m is the mass of the rod, g is the acceleration due to gravity (9.81 m/s^2), and theta is the angle of inclination of the plane.

Express the magnetic force

The magnetic force (Fm) on the rod can be found using the following equation: Fm = I * L * B * sin(phi) Here, I is the current that we need to find, L is the length of the rod, B is the magnetic field strength, and phi is the angle between the direction of the current and the magnetic field. Since the current is along the length of the rod (along the x-axis), and the magnetic field is in the positive y-direction, the angle phi is 90 degrees. Therefore, sin(phi) = 1, and we get: Fm = I * L * B

Equate the gravitational force and magnetic force

In order to hold the rod in place, the magnetic force should balance the gravitational force. Thus, we can equate the two forces and solve for the current, I: I * L * B = m * g * sin(theta) Now, we can solve for the current, I: I = (m * g * sin(theta)) / (L * B) Given the values of m, g, theta, L, and B, we can calculate the magnitude of the current that would have to be passed through the rod to hold it in position on the inclined plane. As for the direction of the current, since the magnetic force should oppose the gravitational force, the current direction should be such that it creates a force in the x-direction opposite to the gravitational force (i.e., upslope) while keeping the rod in position.

Key concepts

Lorentz Force

When charged particles, like electrons in a current-carrying conductor, move through a magnetic field, they experience a force known as the Lorentz force. This force can be described by the equation \( F = q(E + v \times B) \) where \( F \) is the force exerted on the particle, \( q \) is the charge of the particle, \( E \) is the electric field, \( v \) is the velocity of the particle, and \( B \) is the magnetic field. However, for a current-carrying wire, we drop the electric field term and the expression simplifies to \( F = q(v \times B) \), or in more general terms for a wire carrying a current I, the force is \( F = I(L \times B) \).
The direction of the Lorentz force is determined by the right-hand rule, which states that if you point your thumb in the direction of the current and your fingers in the direction of the magnetic field, your palm will face the direction of the force. This force is crucial in many technological applications such as electric motors and generators, and it's the central principle in the exercise of a rod sliding on an inclined plane.

Magnetic Field Strength

The magnetic field strength, denoted by \( B \), represents the magnitude of the magnetic field's force on moving charges or currents within its area of influence. It's typically measured in teslas (T), with Earth's magnetic field being a gentle \( 10^{-5} \) T, while superconducting magnets can produce fields over 10 T. Stronger magnetic fields exert more force on moving charges or currents, which changes the behaviour of charged particles in a material or a current in a wire. In our exercise, the value of \( B \) determines the magnitude of the magnetic force that counteracts the gravitational force on the inclined plane, and figuring out its intensity is essential to calculate the current needed to keep the conducting rod in position.

Gravitational Force

The gravitational force is a force that attracts any objects with mass towards each other. For an object on an inclined plane, like the conducting rod in our scenario, the gravitational force can be represented as \( F_g = m \times g \times \sin(\theta) \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (9.81 m/s² on Earth's surface), and \( \theta \) is the angle of the inclined plane from the horizontal. This force tries to pull the rod down the incline, but in our problem, we are interested in balancing this force with the magnetic force to keep the rod from sliding down.

Current Direction

The current direction is an important factor in determining the direction of the magnetic force experienced by a conductor in a magnetic field, due to the Lorentz force. The established convention is that current direction is the direction in which positive charges would move, even though in most conductors, it's actually electrons (which have a negative charge) that are moving. In our exercise, the correct direction of the current is essential to ensure that the magnetic force counters the gravitational pull. As per the right-hand rule, the current must create a magnetic force upslope to keep the rod stationary on the inclined plane. If the current were to flow in the wrong direction, the resulting magnetic force would either be too weak or could further accelerate the rod's descent.