Figure 29-30 a shows, in cross section, two long, parallel...

Figure $29-30 a$ 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 29-30 $b$ 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 $x_{s}=40.0 \mathrm{~cm}$. (a) At what value of $x>0$ is $B_{y}$ 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 (c) $i_{1}$ and (d) $i_{2}$ ?

Key Concepts

Dependence of Magnetic Field on Distance

The strength of the magnetic field generated by a current-carrying wire decreases with increasing distance from the wire, following an inverse relationship. This distance dependence is vital for calculating the field at various points in space and understanding how the combined fields from multiple sources interact across space.

Magnetic Field of a Current?Carrying Wire

This concept describes how a steady current in a long, straight conductor creates a magnetic field in the space around it. According to Ampère’s law or the Biot–Savart law, the magnetic field at a point depends on the current and the inverse of the distance from the wire. This is a fundamental idea in electromagnetism, forming the basis for analyzing magnetic effects produced by wires and circuits.

Superposition Principle in Magnetism

The superposition principle states that when multiple sources produce magnetic fields, the net magnetic field at any given point is the vector sum of the individual fields produced by each source. This principle is critical when dealing with problems involving more than one current-carrying wire, as it allows the resultant field to be calculated by considering each contribution independently before combining them.

Right?Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. By pointing the thumb of the right hand in the direction of the conventional current, the curled fingers indicate the direction of the magnetic field lines. This rule is essential for visualizing and understanding the orientation of magnetic fields in space.

Optimization of Combined Fields

In many electromagnetic problems, such as finding the maximum net magnetic field at a certain location, the goal is to determine the point where the contributions from different sources combine most constructively. This involves analyzing how the fields vary with distance and direction, allowing one to identify locations where their vector sum is maximized.

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