Question

Same current in same direction is flowing in two parallel wire with 0.06 metre distance apart with each other. The attractive force per metre length of \(3 \times 10^{-3}\) newton is working between the two. Determine the value of current flowing in each wire.

Hint:

For the parallel wires, a helpful way to remember the force direction is "like currents attract, unlike currents repel," which is opposite to the rule for static charges.

Answer 1

Step 1: Understanding the Concept:
Two long, parallel wires carrying currents exert a magnetic force on each other. The magnetic field produced by one wire interacts with the current in the other wire. When the currents are in the same direction, the force is attractive. The magnitude of this force per unit length is given by a standard formula.

Step 2: Key Formula or Approach:
The force per unit length (\(F/L\)) between two parallel wires carrying currents \(I_1\) and \(I_2\) and separated by a distance \(d\) is: \[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} \] In this problem, the currents are the same, so \(I_1 = I_2 = I\). The formula becomes: \[ \frac{F}{L} = \frac{\mu_0 I^2}{2\pi d} \]

Step 3: Detailed Explanation (Calculation):
We are given the following values: \begin{itemize} \item Force per unit length, \(F/L = 3 \times 10^{-3}\) N/m. \item Distance between wires, \(d = 0.06\) m. \item Permeability of free space, \(\mu_0 = 4\pi \times 10^{-7}\) T·m/A. \end{itemize} Substitute these values into the formula and solve for \(I\): \[ 3 \times 10^{-3} = \frac{(4\pi \times 10^{-7}) I^2}{2\pi (0.06)} \] Cancel \(2\pi\) from the numerator and denominator: \[ 3 \times 10^{-3} = \frac{2 \times 10^{-7} \times I^2}{0.06} \] Rearrange to solve for \(I^2\): \[ I^2 = \frac{(3 \times 10^{-3}) \times 0.06}{2 \times 10^{-7}} \] \[ I^2 = \frac{0.18 \times 10^{-3}}{2 \times 10^{-7}} = 0.09 \times 10^4 = 900 \] Take the square root to find the current \(I\): \[ I = \sqrt{900} = 30 \, \text{A} \]

Step 4: Final Answer:
The value of the current flowing in each wire is 30 A.