A wide copper strip of width W is bent to form a tube of...

A wide copper strip of width $W$ is bent to form a tube of radius $R$ with two parallel planar extensions, as shown in Fig. $32-25 .$ There is a current $i$ through the strip, distributed uniformly over its width. In this way a "one-turn solenoid" is formed. (a) Derive an expression for the magnitude of the magnetic field $\vec{B}$ in the FIGURE $32-25=$ Problem 4 .

Key Concepts

Magnetic Field of a Solenoid

A solenoid is a coil of wire designed to produce a magnetic field, typically with the field being most uniform inside the coil. In a 'one-turn solenoid,' even though the geometry is less complex than that of a multi-turn solenoid, the key idea remains to calculate the combined effect of the bent (curved) portion and any straight segments on the magnetic field. The derivation utilizes symmetry and superposition of contributions from different parts of the circuit.

Ampère’s Law

Ampère’s law relates the magnetic field circulating around a closed loop to the net current passing through the loop. This law is especially useful in systems with a high degree of symmetry, such as solenoids or current loops, where one can often simplify the integration to obtain an expression for the magnetic field in a region of interest.

Biot?Savart Law

The Biot?Savart law is the fundamental principle used to calculate the magnetic field produced by a current element. It expresses the contribution to the magnetic field from a differential element of a current?carrying conductor. In problems involving non-standard geometries or finite dimensions, this law is integrated over the current distribution to obtain the total magnetic field at a point in space.

Distributed Current Density

When a current is spread uniformly over a finite conductor, such as a wide strip, the concept of current density becomes important. Current density represents the current per unit cross?sectional area and must be accounted for when integrating the contributions of the current to the magnetic field. This consideration is crucial in deriving an accurate expression for the field in systems where the conductor has a significant spatial extent.

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