Question

A conducting rod of 1 meter length and \(1 \mathrm{~kg}\) mass is suspended by two vertical wires through its ends. An external magnetic field of 2 Tesla 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 \(\left.\mathrm{g}=10 \mathrm{~ms}^{-2}\right]\) (a) \(0.5 \mathrm{Amp}\) (b) \(15 \mathrm{Amp}\) (c) \(5 \mathrm{Amp}\) (d) \(1.5 \mathrm{Amp}\)

Short answer

The current needed to pass through the conducting rod to make the tension in the wires zero is (c) \(5 \mathrm{Amp}\).

Step-by-step solution

Analyze forces acting on the rod

First, let's consider the forces acting on the conducting rod. There are three main forces at play here: 1. Gravitational force (F_g) acting downward due to the mass of the rod 2. Magnetic force (F_m) acting upward, due to the interaction between the current and the magnetic field 3. Tension (T) in the wires supporting the rod at each end. Given that our goal is to make the tension in the wires zero, the magnetic force will have to balance the gravitational force.

Calculate gravitational force

To determine the gravitational force acting on the rod, use the equation: \(F_g = mg \) where m = mass of the rod and g = acceleration due to gravity. In our problem, \(m = 1 kg\) and \(g = 10 m/s^2\). Calculating the gravitational force: \(F_g = (1 kg)(10 m/s^2) = 10 N\)

Determine magnetic force

To find the magnetic force acting on the rod, we will use the equation: \[F_m = BIL\] where B = magnetic field strength, I = current flowing through the rod, and L = length of the rod. In our problem, the magnetic field strength is \(B = 2 T\) and the length of the rod is \(L = 1 m\). We want to find the current I that will make \(F_m = F_g\).

Set up the equation and solve for current I

Now, to make the tension in the wires zero, we want the magnetic force to balance the gravitational force: \[F_g = F_m\] \[mg = BIL\] Plug in the values for m, g, B, and L: \[10 N = (2 T)(1 m)(I)\] Divide both sides of the equation by \(2 T\): \[I = \frac{10 N}{2 T}\] Finally, calculate the current needed to make the tension in the wires zero: \[I = \frac{10 N}{2 T} = 5 A\] Therefore, the correct answer is (c) \(5 \mathrm{Amp}\).

Key concepts

conducting rod

A conducting rod is a component that carries electric current. In this scenario, it is a metallic rod that is hanging vertically, supported by wires. When a current flows through the rod, it interacts with any magnetic fields present. This interaction can result in a force, known as the magnetic force, acting on the rod.
The rod's length and the material it is made from will influence how much current it can carry.

• The rod has a given length, here it is 1 meter, relevant to calculating the magnetic force later.
• The material determines the conductivity, or ease with which current flows through it.

Understanding the properties of the conducting rod is essential for predicting how it behaves under various physical forces.

gravitational force

Gravitational force is the force due to gravity that acts on any object with mass. It pulls objects towards the center of the earth. In the case of the conducting rod, it acts downward. This force is calculated using the equation: \( F_g = mg \) where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity, typically approximately \( 9.81 \, \text{m/s}^2 \), but given as \( 10 \, \text{m/s}^2 \) in this problem for simplicity.
For the conducting rod, with a mass of 1 kg, the calculation becomes: \( F_g = 1 \, \text{kg} \times 10 \, \text{m/s}^2 = 10 \, \text{N} \).

• This force directly affects how the rod will interact with other forces, like the magnetic force.
• Understanding gravitational force helps predict if the rod will remain stationary or move.

magnetic field strength

Magnetic field strength, denoted by \( B \), is a measure of the magnetic influence of a source. It is measured in Teslas (T). In this exercise, the magnetic field strength is given as 2 T. A stronger magnetic field exerts a greater force on current-carrying conductors within it.
This quantity is crucial because it influences the magnetic force exerted on the conducting rod. The relationship is given by: \( F_m = BIL \) where \( I \) is the current through the rod and \( L \) is the length of the rod.

• With a magnetic field strength of 2 T, you can predict how much current is required to produce a force equivalent to the gravitational force.
• This understanding allows the balancing of forces to achieve zero tension in supporting wires.

current and force relationship

The relationship between current and force in the context of magnetic fields is fundamental to electromagnetism. The force exerted on a current-carrying conductor in a magnetic field is given by the equation: \( F_m = BIL \) where \( F_m \) is the magnetic force, \( B \) is the magnetic field strength, \( I \) is the current through the conductor, and \( L \) is the length of the conductor.
In this exercise, we want the force exerted by the magnetic field on the rod to equal the gravitational force to make the tension zero. Solving this for the current \( I \) gives: \( I = \frac{F_g}{BL} \) Using the provided values: \( I = \frac{10 \, \text{N}}{2 \, \text{T} \times 1 \, \text{m}} = 5 \, \text{A} \).

• Understanding this relationship helps in calculating the exact current needed to balance out gravitational forces.
• It forms the basis for many practical applications, including electric motors and generators.