Question

A conducting rod of length \(\ell\) [cross-section is shown] and mass \(\mathrm{m}\) is moving down on a smooth inclined plane of inclination \(\theta\) with constant speed v. A vertically upward mag. field \(\mathrm{B}^{-}\) exists in upward direction. The magnitude of mag. field \(B^{-}\) is(a) \([(\mathrm{mg} \sin \theta) /(\mathrm{I} \ell)]\) (b) \([(\mathrm{mg} \cos \theta) /(\mathrm{I} \ell)]\) (c) \([(\mathrm{mg} \tan \theta) /(\mathrm{I} \ell)]\) (d) \([(\mathrm{mg}) /(\mathrm{I} \ell \sin \theta)]\)

Short answer

The magnitude of the magnetic field B is given by: \( B = \frac{mg\sin\theta}{I\ell} \)

Step-by-step solution

Calculate gravitational force acting on the rod

The gravitational force acting on the rod can be resolved into two components: one perpendicular to the inclined plane (mgcosθ) and one parallel to it (mgsinθ). The force parallel to the inclined plane (mgsinθ) causes the rod to slide down.

Calculate the magnetic force acting on the rod

According to Faraday's law of electromagnetic induction, the conducting rod experiences an induced electromotive force (EMF) as it moves in the magnetic field B. This induces an electric current, I, through the rod. Due to the magnetic field, this electric current exerts a magnetic force on the rod. The magnetic force, F, acting on the rod can be given by the equation: \( F = I\ell B \) where F is the magnetic force, I is the induced current, 𝑙 is the length of the rod, and B is the magnetic field.

Apply equilibrium conditions for the rod moving with constant speed

Since the rod is moving at a constant speed, the net force acting on the rod along the direction of motion is zero. Therefore, we have: Governing_force (Gravity) = Governing_force (Magnetic) or \( mgsin\theta = I \ell B \)

Solve for the magnetic field, B

Now, we will rearrange the equation derived in step 3 to find B: \( B = \frac{mgsin\theta}{I\ell} \) Comparing the given options with our derived expression for B, we can see that the correct answer is given by option (a): \( B = \frac{mg\sin\theta}{I\ell} \)

Key concepts

Electromagnetic Induction

Electromagnetic induction is a fascinating concept in physics that refers to the process where a change in magnetic field induces an electromotive force (EMF) across a conductor. Imagine a conducting rod moving through a uniform magnetic field. As it moves, this change in its position relative to the magnetic field produces an EMF, which in turn can drive an electric current through the conductor. This phenomenon was first discovered by Michael Faraday, and it forms the basis for many electrical technologies, including generators and transformers. The EMF induced can be calculated using Faraday's Law of Induction, which states that the induced EMF in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. In formula terms:- \( \text{EMF} = - \frac{d\Phi}{dt} \)Where \( \Phi \) is the magnetic flux. This principle helps to explain how motion through a magnetic field translates into electrical energy.

Gravitational Force on Inclined Plane

Gravitational force on an inclined plane is a common topic in physics that involves understanding how gravitational force acts on bodies placed on an inclined surface. When a body is on an inclined plane, gravity acts directly downwards, but this force can be resolved into two components:

• Parallel to the plane: This is the component that causes the body to slide down the plane. It is given by \( mg \sin \theta \), where \( m \) is mass of the body and \( \theta \) is the angle of inclination.
• Perpendicular to the plane: This component presses the body into the plane, and is given by \( mg \cos \theta \).

Understanding these components is crucial when analyzing systems on inclined planes, such as this exercise where balance between gravitational and magnetic forces allows the rod to move at a constant speed. Resolving forces helps in understanding how objects will accelerate or remain in equilibrium on slopes.

Magnetic Force Calculation

Calculating magnetic force is a vital step in understanding how magnetic fields interact with moving charges, such as in this rod on an inclined plane. When a conductor like our rod is within a magnetic field, and it carries a current, a magnetic force is exerted on it. The magnitude of this force can be calculated using the formula:\[ F = I \ell B \]Where:

• \( F \) is the magnetic force.
• \( I \) is the current flowing through the conductor.
• \( \ell \) is the length of the conductor within the magnetic field.
• \( B \) is the magnetic field strength.

This force is perpendicular to both the direction of the current and the magnetic field, leading to what is often measured as Lorentz force. Understanding this calculation is essential in questions involving electromagnetic systems, where the balance between electrical and magnetic forces determines the motion of the system.

Equilibrium in Physics

In physics, equilibrium refers to a condition where the net force or net moment on a system is zero, leading to the system being in a state of rest or moving with constant velocity. For the rod in the given exercise, achieving equilibrium means that the downward gravitational force and upward magnetic force must balance each other as the rod moves down the incline. This is evident in the relationship:- \( mg \sin \theta = I \ell B \)Where the gravitational force \( mg \sin \theta \) is matched by the magnetic force \( I \ell B \). This concept of equilibrium is crucial in physics, as it helps predict how systems behave when subjected to various forces. It is the reason why our rod moves at a constant speed down the incline despite being in a gravitational field and interacting with a magnetic field.Equilibrium conditions are central in mechanics and electromagnetism, allowing us to make precise predictions and understand complex interactions between forces in static and dynamic systems.