Figure 29-50 a shows, in cross section, two long, parallel...

Figure $29-50 a$ shows, in cross section, two long, parallel wires carrying current and separated by distance $L .$ The ratio $i_{1} / i_{2}$ of their currents is $4.00 ;$ the directions of the currents are not indicated. Figure $29-50 b$ shows the $y$ component $B_{y}$ of their net magnetic field along the $x$ axis to the right of wire $2 .$ The vertical scale is set by $B_{y s}=4.0 \mathrm{nT}$ , and the horizontal scale is set by $x_{s}=20.0 \mathrm{cm}$ . (a) At what value of $x>0$ is $B_{v}$ maximum? (b) If $i_{2}=3 \mathrm{mA}$ , what is the value of that maximum? What is the direction (into or out of the page $)$ of $(\mathrm{c}) i_{1}$ and $(\mathrm{d}) i_{2}$ ?

Figure $29-50 a$ shows, in cross section, two long, parallel wires carrying current and separated by distance $L .$ The ratio $i_{1} / i_{2}$
of their currents is $4.00 ;$ the directions of the currents are not indicated. Figure $29-50 b$ shows the $y$ component $B_{y}$ of their net magnetic field along the $x$ axis to the right of wire $2 .$ The vertical scale is set by $B_{y s}=4.0 \mathrm{nT}$ , and the horizontal scale is set by $x_{s}=20.0 \mathrm{cm}$ .
(a) At what value of $x>0$ is $B_{v}$ maximum? (b) If $i_{2}=3 \mathrm{mA}$ , what is the value of that maximum? What is the direction (into or out of the
page $)$ of $(\mathrm{c}) i_{1}$ and $(\mathrm{d}) i_{2}$ ?

Key Concepts

Magnetic Field of a Long Straight Wire

The magnetic field produced by a long straight current-carrying wire is calculated using Ampère’s Law or the Biot–Savart Law, which shows that the field’s magnitude is proportional to the current and inversely proportional to the distance from the wire. This concept underpins the analysis of magnetic interactions in systems with one or more current-carrying elements.

Superposition Principle

When multiple magnetic fields are present, the net magnetic field at any point is the vector sum of the fields produced by each source. This principle is essential in solving problems where the effects of different currents or magnets overlap.

Right-Hand Rule

The right-hand rule is a method used to determine the direction of the magnetic field around a current-carrying conductor. By aligning your thumb in the direction of the current, the curl of your fingers shows the circular direction of the magnetic field around the wire.

Optimization of Magnetic Field Strength

Determining the location where the magnetic field is maximized, or minimized, involves analyzing the spatial variation of the individual fields and finding the conditions under which they constructively or destructively interfere. This approach often requires setting up equations based on field expressions and solving for the points of interest.

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