Question

Two infinitely long straight parallel conductors carrying currents \(I_1\) and \(I_2\) are held at a distance d apart in vacuum. The force F on a length L of one of the conductors due to the other is _______.
Fill in the blank with the correct answer from the options given below

• A. proportional to L but independent of \( I_1\times I_2\)
• B. proportional to \(I_1 × I_2\) but independent of length L
• C. proportional to \(I_1 × I_2 × L\)
• D. proportional to\( \frac{L}{ I_1 × I_2}\)

Answer 1

The Correct Option is C

To determine the force between two parallel conductors carrying currents, we can use the equation derived from Ampère's Law and the Biot-Savart Law. The force per unit length f between two parallel currents I₁ and I₂ separated by a distance d in a vacuum is given by:

\( f = \frac{\mu_0}{2\pi} \cdot \frac{I_1I_2}{d} \)

where μ₀ is the permeability of free space. The total force F on a length L of one conductor is:

\( F = f \cdot L = \left(\frac{\mu_0}{2\pi} \cdot \frac{I_1I_2}{d}\right) \cdot L \)

From this equation, it is clear that the force F is proportional to the product I₁ × I₂ × L.

Therefore, the correct answer is that the force on a length L of one of the conductors is proportional to \(I_1 \times I_2 \times L\).

Answer 2

The force (F) on a length (L) of one conductor due to the other is given by the formula:

F = (μ₀ * I₁ * I₂ * L) / (2πd)

Where:

• μ₀ is the permeability of free space
• I₁ and I₂ are the currents in the conductors
• L is the length of the conductor
• d is the distance between the conductors

From this formula, we can see that the force (F) is:

• Proportional to the product of the currents (I₁ * I₂).
• Proportional to the length (L) of the conductor.
• Inversely proportional to the distance (d) between the conductors.

Therefore, the force F is proportional to I₁ * I₂ * L.

The correct answer is:

Option 3: proportional to I₁ × I₂ × L