Most introductory physics courses derive the energy densities of the static electric and magnetic field for the simple cases of parallel plate capacitors and infinitely long solenoids. Authors then inform readers, usually without proof, that the energy density equations derived for the simple cases are actually the correct equations for all capacitors and inductors, regardless of their shape. The proof of the general case is typically omitted because it involves the differential calculus of vector fields. In view of this, we provide a derivation for the energy density only based on integral calculus for capacitors and inductors of any kind. The derivation, albeit seemingly complicated at first, is conceptually simple enough for introductory physics courses and does not require any knowledge of the differential calculus of vector fields.

Topics
Inductors, Capacitors, Electromagnetic coils, Magnetic energy, Maxwell equations, Vector fields
1.
Ronald L.
Reese
,
University Physics
(
Brooks
/
Cole Publishing Company
,
2000
), pp.
819
–
820
and pp. 979–980.
Google Scholar
 
2.
Raymond A.
Serway
 and
John W.
Jewett
, Jr.,
Physics for Scientists and Engineers
(
Brooks
/
Cole Publishing Company
,
2004
), pp.
807
–
808
and pp. 1011–1012.
Google Scholar
 
3.
Hugh D.
Young
,
Roger A.
Freedman
, and
A. Lewis
Ford
,
Sears and Zemansky's University Physics: With Modern Physics
, 13th ed. (
Addison-Wesley
,
2012
), pp.
796
–
798
and pp. 998–1000.
Google Scholar
 
4.
David
Halliday
,
Robert
Resnick
, and
Jearl
Walker
,
Fundamentals of Physics, Extended
, 11th ed. (
Wiley
,
2018
), pp.
728
–
730
and pp. 887–889.
Google Scholar
 
5.
Debora
Katz
,
Physics for Scientists and Engineers: Foundations and Connections With Modern Physics
(
Cengage Learning
,
2017
), Secs. 27-2 and 33-4.
Google Scholar
 
6.
Ramamurti
Shankar
,
Fundamentals of Physics II
(
Yale U. P
.,
2016
), Secs. 7.1 and 11.6.
Google Scholar
 
7.
Richard P.
Feynman
,
Robert B.
Leighton
, and
Mattew
Sands
,
The Feynman Lectures on Physics: Definitive Edition
(
Addison-Wesley
,
2006
), Vol.
II
, pp.
8
–
11
and pp. 17–15.
Google Scholar
 
8.
David J.
Griffiths
,
Introduction to Electrodynamics
, 4th ed.(
Cambridge U. P
.,
2017
), pp.
94
–
95
and pp. 299–301.
Google Scholar
Crossref
Search ADS
 
9.
Andrew
Zangwill
,
Modern Electrodynamics
(
Cambridge U. P
.,
Cambridge
,
2012
), p.
78
and p. 385.
Google Scholar
Crossref
Search ADS
 
10.
John D.
Jackson
,
Classical Electrodynamics
, 3rd ed. (
John Wiley & Sons, Inc
.,
1999
), p.
41
and p. 213.
Google Scholar
 
11.
The existence of “equipotential” surfaces for the magnetic flux calculation is guaranteed by Maxwell's equations for the static fields in regions free of electric charges (empty space), stationary or moving, ∇·E=0,∇×E=0,∇·B=0,∇×B=0.These equations allow for the existence of scalar potentials ϕ and ψ such that E=−∇ϕ and B=−∇ψ for the static electric and magnetic field E⁠, B⁠, respectively.8,10,12 The surfaces formed by ϕ= constant, ψ= constant are the “equipotential” surfaces to which the local field is everywhere normal. Although obeying the same differential laws, being orthogonal to the local fields is the only property shared by the equipotential surfaces of the electric and magnetic fields. Globally, they have different behaviors (topology) due to the boundary conditions that the fields must satisfy and the topology of the space in which they exist.8,10,12
12.
Julius A.
Stratton
,
Electromagnetic Theory
(
McGraw-Hill Book Company
,
1941
), p.
227
.
Google Scholar
 
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