Amount of magnetic energy stored in a cylinder

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SUMMARY

The discussion focuses on calculating the energy density stored in a magnetic field using the formula \( u = \frac{B^2}{2\mu_0} \). The participant initially used Ampere's law to determine the magnetic field strength but struggled to integrate the results to find the energy density. It was clarified that this formula is applicable in all cases, including in vacuum and air, and that the variation of field strength depends on the geometry of the conductors involved.

PREREQUISITES
  • Understanding of Ampere's Law
  • Familiarity with inductance and the formula \( L = \frac{\text{Flux}}{\text{Current}} \)
  • Knowledge of magnetic field strength and energy density concepts
  • Basic calculus for integration
NEXT STEPS
  • Research the derivation of the energy density formula \( u = \frac{B^2}{2\mu_0} \)
  • Study the applications of Ampere's Law in different geometries
  • Explore the behavior of magnetic fields in various materials beyond vacuum and air
  • Learn about the implications of magnetic field strength variations in practical applications
USEFUL FOR

Physicists, electrical engineers, and students studying electromagnetism who seek to understand the principles of magnetic energy storage and its applications in various contexts.

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Homework Statement
Find the amount of energy stored in thin cylinder of length l,around a conductor carrying current I as shown.

The figure is in Attempt at a solution.
Relevant Equations
E=0.5LI^2
Ampere's Circuital Law
Inductance=Flux/Current
1565245344848-96805623.jpg


I first calculated Magnetic field at a distance x using Ampere's law.Then i took an element and wrote its inductance using L=Flux/Current.
Then i integrated from a to b.
But i arrived at nothing.
Help.
 
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What is the energy density stored in a magnetic field of a given strength?
 
mfb said:
What is the energy density stored in a magnetic field of a given strength?
It is Energy per unit volume,so should i first find energy for a part by energy density and integrate it?
 
You should find a relation that links energy density of a magnetic field (what you need) to the field strength (what you calculated).
 
mfb said:
You should find a relation that links energy density of a magnetic field (what you need) to the field strength (what you calculated).
yeah i think it is energy density=B2##/##2u0
 
mfb said:
You should find a relation that links energy density of a magnetic field (what you need) to the field strength (what you calculated).
Thanks i got the answer.Earlier i thought that this result was valid for only solenoid.But it is for all cases is it?
 
The formula for the energy density is valid for all cases (in vacuum, technically, but it is a good approximation in air). How the field strength varies in space depends on the geometry of the cables.
 

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