Figure 29-35 a shows an ele- f ment of length d s=1.00 μm in a -ϑ very long straight wire carryi

step 1 Hi, in the given problem, as per biote and sever's law, the small magnetic field due to a curren

Figure 29-35 a shows an ele- f ment of length d s=1.00 μm in...

Figure $29-35 a$ shows an ele- $f$ ment of length $d s=1.00 \mu \mathrm{m}$ in a $\stackrel{-}{\vartheta}$ very long straight wire carrying $\vartheta$ current. The current in that element sets up a differential magnetic field $d \vec{B}$ at points in the $(b)$ surrounding space. Figure $29-35 b$ gives the magnitude $d B$ of the field $\quad$ Figure 29-35 Problem $2 .$ for points $2.5 \mathrm{~cm}$ from the element, as a function of angle $\theta$ between the wire and a straight line to the point. The vertical scale is set by $d B_{s}=60.0 \mathrm{pT}$. What is the magnitude of the magnetic field set up by the entire wire at perpendicular distance $2.5 \mathrm{~cm}$ from the wire?

Key Concepts

Biot-Savart Law

The Biot-Savart law relates the magnetic field produced at a point by a small element of current to the magnitude of the current, the length and orientation of the current element, and the distance from the element to the point. It provides the fundamental basis for calculating the contribution to the magnetic field (dB) from each differential segment of the current-carrying conductor.

Superposition Principle

In electromagnetism, the superposition principle states that the total magnetic field created by multiple sources is the vector sum of the fields produced by each source individually. This principle allows for the integration of the differential magnetic fields from each current element along the wire to obtain the overall magnetic field at a point.

Integration Over a Current Distribution

Integration is used to sum the contributions of the differential fields generated by each element of the current-carrying wire. By integrating the effects of all the infinitesimal current elements, one can determine the complete magnetic field at a specific location in the space surrounding the wire.

Magnetic Field of a Long Straight Wire

For a long, straight current-carrying wire, the magnetic field at any point a perpendicular distance away from the wire is given by a simplified relation (often derived using Ampère's law) where the field magnitude is inversely proportional to the distance from the wire. This concept is essential for understanding how the integrated contributions reduce to a familiar and simple analytical form.

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