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) \(5 \mathrm{Amp}\).