A solenoid has the following dimensions: diameter =5.0 cm n=45 turns / cm, length =23 cm (a) W

step 1 In terms to the electromagnetic induction in which we have a solenoid having diameter D equals t

A solenoid has the following dimensions: diameter =5.0 cm...

A solenoid has the following dimensions: diameter $=5.0 \mathrm{~cm}$ $n=45$ turns $/ \mathrm{cm},$ length $=23 \mathrm{~cm}$
(a) What's its inductance?
(b) Find the current when the solenoid stores $0.50 \mathrm{~J}$ of energy.

Key Concepts

Inductance

Inductance is a measure of a circuit element’s ability to store energy in its magnetic field when current flows through it. For coils and solenoids, it quantifies the opposition to changes in current, with a higher inductance indicating that more energy is stored for a given current change.

Solenoid Geometry and Turn Density

The physical dimensions of a solenoid—its length, diameter, and the density of its turns—are crucial in determining its inductance. The cross-sectional area and the total number of turns (which is the turn density multiplied by the length) directly influence the strength and distribution of the magnetic field within the solenoid.

Magnetic Energy Storage

An inductor stores energy in the magnetic field created by the flow of current through it. The stored energy is given by the formula U = 1/2 L I^2, where U is the energy, L is the inductance, and I is the current. This relationship is fundamental in analyzing the behavior of circuits during transient events and energy transfer processes.

Relationship Between Current, Inductance, and Energy

The interplay between current, inductance, and stored energy allows one to calculate one quantity when the other two are known. For example, given the energy stored in an inductor and its inductance, one can solve for the current, highlighting the dependency between these electrical parameters.

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