Problem 43 - Master Resource Book in JEE Main Physics

Question

A metal wire of mass \(m\) slides without friction on two rails placed at a distance \(l\) apart. The track lies in a uniform vertical magnetic field \(B\). A constant current \(i\) flows along the rails across the wire and brack down the other rail. The acceleration of the wire is (a) \(\frac{B m i}{l}\) (b) \(\mathrm{mBi} l\) (c) \(\frac{B i l}{m}\) (d) \(\frac{\mathrm{mil}}{B}\)

Step-by-Step Solution

Answer

The correct answer is (c) \( \frac{B i l}{m} \).

Step 1: Understanding the situation

A wire moves on two rails in a uniform magnetic field, with current flowing through the circuit consisting of the wire and rails.

Step 2: Analyzing the Force

The magnetic force, known as the Lorentz force, acting on the wire is given by the formula \( F = i L B \), where \( i \) is the current, \( L \) the length of the wire (distance between rails), and \( B \) the magnetic field.

Step 3: Applying Newton's Second Law

According to Newton's second law, the net force acting on an object is equal to the product of its mass \( m \) and acceleration \( a \). So, we write: \( F = m a \).

Step 4: Solving for Acceleration

Substituting the expression for the magnetic force from Step 2 into Newton's second law from Step 3, we have: \( i L B = m a \). Rearranging to solve for \( a \), we get \( a = \frac{i L B}{m} \).

Step 5: Identifying the Correct Option

From our calculations, the acceleration \( a = \frac{i L B}{m} \). Comparing options, the correct answer is option (c) \( \frac{B i l}{m} \).

Key Concepts

Lorentz Force

The Lorentz Force is a fundamental concept in electromagnetism that describes the force exerted on a charged particle moving through a magnetic field. When a current flows through a conductor in a magnetic field, as seen in our exercise, the conductor experiences this force. This force is given by the equation:\[ F = i L B \]Where:

• i is the current flowing through the conductor (wire).


• L is the length of the conductor.


• B is the magnetic field strength.

The direction of the Lorentz Force can be determined using the right hand rule: point your thumb in the direction of the current, your fingers in the direction of the magnetic field, and your palm will face in the direction of the force exerted on the wire. This force causes the wire to accelerate along the rails. Understanding this principle is key to solving problems involving interactions between conductors and magnetic fields.

Newton's Second Law

Newton's Second Law of Motion is a critical principle in physics. It states that the acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass. The mathematical expression for this law is:\[ F = m a \]Where:

• F is the net force acting on the object.


• m is the mass of the object.


• a is the acceleration of the object.

In our exercise, this law helps us relate the magnetic force (Lorentz Force) acting on the wire to its acceleration. By equating the Lorentz force to the product of the mass and acceleration of the wire, we can solve for the acceleration. This gives:\[ i L B = m a \]Rearranging this equation, we find the acceleration:\[ a = \frac{i L B}{m} \]This step is crucial in determining how and why the wire accelerates under the influence of the magnetic field.

Magnetic Field

A magnetic field is a vector field that represents the magnetic influence on moving electric charges, electric currents, and magnetic materials. In our situation, the magnetic field is uniform and vertical, influencing the current-carrying wire that slides along the rails. The strength of the magnetic field is denoted by the symbol B. It is a crucial parameter in calculating the Lorentz force; the stronger the field, the greater the force on a current-carrying conductor. The orientation of the magnetic field also plays a role in the direction of the force, which is perpendicular to both the direction of the current and the magnetic field. Understanding the properties and effects of magnetic fields is essential in electromagnetism and is widely applied in technologies such as electric motors and generators, where magnetic fields are used to induce movement and generate electricity.

Current in Conductors

The flow of electric charge through a conductor is known as electric current, and it is a key factor in the operation of electrical circuits and devices. In the context of our exercise, the wire on the rails acts as a conductor through which the current i is flowing. As the current flows, it interacts with the magnetic field produced in the region. This interaction is what causes the Lorentz force to act on the conductor, resulting in its acceleration. Current is measured in amperes (A), and its effect in this scenario is directly proportional to the force experienced by the conductor. It's important to recognize the role of current in the electronic circuits:

• It influences the magnitude of forces in electromagnetic environments.


• Its direction is significant in determining the resulting Lorentz force direction using the right-hand rule.

Understanding current flow is crucial not only for physics problems but also for practical applications in electronics and electrical engineering.