Figure 29-85 shows, in cross section, two long parallel...

Figure $29-85$ shows, in cross section, two long parallel wires that are separated by distance $d=18.6 \mathrm{cm} .$ Each carries $4.23 \mathrm{A},$ out of the page in wire 1 and into the page in wire $2 .$ In unit-vector notation, what is the net magnetic field at point $P$ at distance $R=34.2 \mathrm{cm},$ due to the two currents?

Figure $29-85$ shows, in cross section, two long parallel wires that are
separated by distance $d=18.6 \mathrm{cm} .$
Each carries $4.23 \mathrm{A},$ out of the page
in wire 1 and into the page in wire $2 .$
In unit-vector notation, what is the
net magnetic field at point $P$ at distance $R=34.2 \mathrm{cm},$ due to the two
currents?

Key Concepts

Magnetic Field of a Long Straight Conductor

A long, current-carrying wire produces a magnetic field that circulates around the wire. Its magnitude at a distance r from the wire is given by B = ??I/(2?r), where ?? is the permeability of free space and I is the current. This relationship is derived from the Biot–Savart law or Ampère’s law and is foundational when analyzing magnetic fields in simple geometries.

Right-Hand Rule

The right-hand rule is a mnemonic for determining the direction of the magnetic field created by a current flow. By pointing the thumb of the right hand in the direction of the conventional current, the curl of the fingers shows the circular path of the magnetic field lines around the wire. This concept is crucial when figuring out the vector nature of individual magnetic field contributions.

Superposition Principle

The superposition principle states that when multiple magnetic fields are present, the net magnetic field at any point is the vector sum of the individual fields. This principle allows for the independent calculation of magnetic fields from different current sources, which can then be added together to determine the overall effect at a specific location.

Unit Vector Notation

Unit vector notation is a method to express vectors in terms of their components along the coordinate axes using unit vectors (for example, î, ?, and k?). This notation facilitates the clear representation of the magnitude and direction of vector quantities like the magnetic field, especially when performing vector addition for fields from multiple sources.

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