Problem 63 - Master Resource Book in JEE Main Physics

Question

A conducting rod of \(1 \mathrm{~m}\) length and \(1 \mathrm{~kg}\) mass is suspended by two vertical wires through the ends. An external magnetic fields of \(2 \mathrm{~T}\) is applied normal to the rod. Now the current to be passed through the rod so as to make the tension in the wires zero is (Take \(\left.g=10 \mathrm{~ms}^{-2}\right) \quad[\) Kerala CET 2007] (a) \(0.5 \mathrm{~A}\) (b) \(15 \mathrm{~A}\) (c) \(5 \mathrm{~A}\) (d) \(1.5 \mathrm{~A}\) (e) \(15 \mathrm{~A}\)

Step-by-Step Solution

Answer

The current required is 5 A (option c).

Step 1: Understand the Problem

We have a conducting rod of length 1 m and mass 1 kg. It is suspended by two wires, and a magnetic field of 2 T is applied perpendicularly to the rod. We need to find the current required to make the tension in the wires zero.

Step 2: Analyze the Forces

The gravitational force acting on the rod is equal to its weight, which is given by the equation: \[ F_{ ext{gravity}} = mg \]where \( m = 1 \) kg and \( g = 10 \) m/s². Calculate \( F_{ ext{gravity}}. \)

Step 3: Calculate the Gravitational Force

Using the formula for gravitational force:\[ F_{ ext{gravity}} = 1 \times 10 = 10 \text{ N} \]The gravitational force on the rod is 10 N.

Step 4: Determine the Magnetic Force Needed

To make the tension in the wires zero, the magnetic force must be equal and opposite to the gravitational force, i.e., 10 N upwards. The magnetic force on the rod can be calculated by:\[ F_{ ext{magnetic}} = BIL \]where \( B = 2 \) T, \( L = 1 \) m, and \( I \) is the current flowing through the rod.

Step 5: Solve for Current

Set the magnetic force equal to the gravitational force:\[ BIL = mg \]Substitute the known values:\[ 2 \times I \times 1 = 10 \]Solve for \( I \):\[ I = \frac{10}{2} = 5 \text{ A} \]

Step 6: Verify the Solution

The calculated current of 5 A produces a magnetic force of 10 N, which exactly balances the gravitational force, making the tension in the wires zero.

Key Concepts

Current Calculation

In this scenario, calculating the current involves understanding the forces at play. The rod in the exercise experiences gravity pulling it downwards. To counter this, a magnetic force generated by electric current must act upwards.
The relationship governing the magnetic force in the presence of a magnetic field is given by the formula:

• Magnetic Force (\( F_{\text{magnetic}} \) = \( BIL \), where:
  • \( B \) is the magnetic field in tesla (T),
  • \( I \) is the current in amperes (A),
  • \( L \) is the length of the rod in meters (m).

To find the current that balances out the gravitational pull, you set this magnetic force equal to the gravitational force acting on the rod. Solving this gives us the current needed to ensure no tension in the wires.

Tension in Wires

The primary goal of this exercise is to make the tension in the wires zero. Tension arises due to the weight of the rod pulling downwards. If this weight can be exactly countered by an upward force, the tension is nullified.
In simple terms, tension is the force you would feel if you were to hold one end of the wire while the rod hung from the other. To reduce the tension to zero:

• The upward magnetic force must equal the downward gravitational force.
• When these two forces balance perfectly, the wires no longer feel the weight of the rod.

This state is achieved when the current calculated previously flows through the rod, generating exactly enough magnetic force upwards to counteract gravity.

Gravitational Force

Gravitational force is the natural force exerted by the earth that pulls objects towards its center. For the rod to hover freely without tension in the supporting wires, the force pulling it downward (gravitational force) must be clearly understood and countered.
The gravitational force exerted on an object is calculated using the formula:

• \( F_{\text{gravity}} = mg \), where:
  • \( m \) is the mass of the object (in kg),
  • \( g \) is the acceleration due to gravity (approximately 10 \( m/s^2 \) for this problem).

In this exercise, the gravitational force turns out to be 10 N. This force needs to be countered by an equivalent magnetic force for the wires to experience zero tension.

Magnetic Field Influence

A magnetic field has significant influence on a current-carrying conductor. In this exercise, when the magnetic field is perpendicular to the current, it exerts a lateral force on the rod, known as the magnetic force.
This magnetic force can be harnessed to counteract other forces like gravity. Here's how it occurs:

• The conductor experiences a force due to the interaction of the current and the magnetic field.
• The magnitude of this force is determined by the formula \( F_{\text{magnetic}} = BIL \). This shows that the force is directly proportional to the magnetic field strength, the current, and the length of the conductor.

With a magnetic field strength of 2 T, an effective combination of current through the rod not only counters gravity but eliminates tension in the rod's supporting wires, showcasing the powerful influence a magnetic field can have on electrical conductors.