A closely wound, circular coil with radius 2.40 cm has 800 turns. a) What must the current in th

step 1 Consider a closely wound circular coil which has a radius of 2 .40 cm and 800 tons. First we nee

A closely wound, circular coil with radius 2.40 cm has 800...

A closely wound, circular coil with radius 2.40 $\mathrm{cm}$ has 800 turns. a) What must the current in the coil be if the magnetic field at the center of the coil is 0.0580 $\mathrm{T}$ ? b) At what distance $x$ from the center of the coil, on the axis of the coil, is the magnetic field half its value at the center?

Key Concepts

Magnetic Field at the Center of a Current Loop

This concept involves calculating the magnetic field produced at the center of a single current-carrying circular loop using the Biot?Savart law. The field is directly proportional to the current and inversely proportional to the radius of the loop. Understanding this simplifies the analysis of simple coil systems and is foundational for more complex configurations.

Magnetic Field along the Axis of a Current Loop

Extending the analysis beyond the center, this concept describes how the magnetic field varies along the axis of a circular current loop. The field expression involves the distance from the center and shows that the field strength decreases with distance. This is crucial for determining positions in space where the field reaches specific values, such as half its central value.

Effect of Multiple Turns in a Coil

When a coil consists of multiple closely wound turns, the magnetic field contributions from each loop add together linearly. This means that the overall magnetic field is multiplied by the number of turns, enhancing the field strength provided the coils are arranged such that their fields reinforce each other.

Permeability of Free Space

This is a physical constant that appears in the equations governing magnetic fields produced by current-carrying conductors. It quantifies how a magnetic field propagates in vacuum and is essential in relating the current and geometry of a coil to the resulting magnetic field.

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