That was a really good explanation of what was happening for the forces of the perpendicular lines, especially with the left and right hand rules being applied to the lines. Something that I was curious about was the conclusion that you drew in the last line: “Nevertheless, they do not produce a net torque on one another” if my understanding is correct, then a force could indicate torque. Why is there no torque produced in this scenario?

Answer

Why No Torque Between Perpendicular Current-Carrying Wires?

🔧 Why Is There No Torque Between Perpendicular Current-Carrying Wires?

🔍 Understanding the Scenario

Consider two wires carrying currents that are perpendicular to each other. One wire lies along the x-axis (let’s call it Wire A) and the other along the y-axis (Wire B). Both currents generate magnetic fields, and these fields exert magnetic forces on the other wire.

You might expect that these forces would create a torque since they act at different locations and directions. However, as counterintuitive as it seems, the net torque between these wires is actually zero. Here’s why.

🧲 Magnetic Force and Torque: Key Concepts

Magnetic Force: A current-carrying wire in a magnetic field experiences a force given by the Lorentz force law: F = I (L × B).
Torque: Torque (τ) is the tendency of a force to cause rotation. It is defined as τ = r × F, where r is the position vector from the axis of rotation to the point of force application.

      ✅ Torque depends not only on the force but also on the location and direction of that force.
    

🧠 Analyzing the Forces Between the Wires

Each wire creates a magnetic field that circles around it. According to the right-hand rule:

Wire A’s field circulates in the yz-plane.
Wire B’s field circulates in the xz-plane.

These fields interact with the other wire’s current and exert forces on them. These forces are perpendicular to both the direction of current and magnetic field.

      ⚠️ But the key point is: The forces are balanced in such a way that they do not create a rotational effect about any axis.
    

🧭 Why Is the Net Torque Zero?

Let’s break it down:

Each wire exerts a force on the other.
The location of these forces may be offset, but they are symmetric in nature.
Because these forces are equal in magnitude and opposite in direction, they cancel out any potential rotational effect.

🧩 Even though a force can cause a torque, it only does so if it has a lever arm (distance from axis of rotation). In this case, the geometry and symmetry result in a zero net moment (torque).

🔄 Summary

🔌 Two perpendicular wires carrying current exert forces on each other.
🧲 These forces arise from magnetic interactions.
🌀 Despite the forces, the net torque is zero because the configuration is symmetrical and forces cancel each other rotationally.
📏 This aligns with Newton’s third law and principles of rotational equilibrium.

      ✅ Final Insight: Yes, a force can produce torque—but only if it acts off-center in a way that causes rotation. In this case, the magnetic forces balance in such a way that no net torque results.
    

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