91Ó°ÊÓ

Understanding the concept of current in a wire is crucial when studying magnetic fields. An electric current is essentially the flow of electric charge through a conductor. In the context of wires, we usually speak about the

• amount of charge flowing
• how fast it is moving

which is measured in amperes (A). For example, wire 1 in our exercise carries a current of 6.5 A. This means a significant flow of electric charges is moving through the wire each second.

In a long, straight wire, this moving charge creates a magnetic field around it. This magnetic field spins around the wire, resembling concentric circles. The strength of this field can be calculated using the formula: \[ B = \frac{\mu_0 I}{2 \pi r} \]where:

• \(B\) is the magnetic field
• \(\mu_0\) is the permeability of free space
• \(I\) is the current in amperes
• \(r\) is the distance from the wire to the point where the magnetic field is measured

The right-hand rule is a simple yet powerful tool that helps us visualize the direction of the magnetic field created by a current. This rule states that if you curl the fingers of your right hand in the direction of the current flow in the wire, your thumb will point in the direction of the magnetic field.

For our exercise, we're looking at two wires: one with a current going into the page. If we apply the right-hand rule here,

• curling our fingers into the page aligns them along the circles of the magnetic field around wire 1.
• Your thumb will point towards the observed magnetic field's direction.

This rule helps us determine that the field at point P is active and in a direction that depends on the current's flow in the wires. When determining the direction of the magnetic field contributed by wire 2 to counteract wire 1's field, the right hand's orientation will be opposite, considering it needs to balance out.

The superposition principle allows us to calculate the net magnetic field at a point by adding together the individual magnetic fields from each current. This is immensely helpful when multiple currents, like in our exercise, influence a point in space.

At point P, we want the net magnetic field to be zero, which involves

• calculating the magnetic field from wire 1
• understanding the requirement for wire 2's field to counter wire 1 precisely

With superposition, the idea is that these fields simply add up: their directions and magnitudes directly combined.

For the fields to cancel out at point P, wire 2’s field should be equal in magnitude but opposite in direction to that of wire 1. Using the magnetic field formula for our specific contexts, we set up the situation such that \[ \frac{\mu_0 I_2}{2\pi d_2} + \left(-\frac{\mu_0 \times 6.5}{2\pi \times 0.0225}\right)=0 \]. Using this approach, we solved for the current needed in wire 2. Superposition helps simplify complex electromagnetic interactions by focusing on balance and equivalence.