Danish Scien i ic Jou nal No100, 2025 55
RESEARCH ON THE APPLICATION OF SYMMETRY METHODS IN SOLVING UNIVERSITY
PHYSICS PROBLEMS — TAKING MECHANICS, ELECTROSTATIC FIELDS AND STEADY
MAGNETIC FIELDS AS EXAMPLES
Dong Yongsheng
JiNing No mal Uni e si y, Science and Technology Depa men ,
Ulanqab, Inne Mongolia, 012000 ,P. R. China
Fund P ojec : Key P ojec o Na u al Science Resea ch o Jining No mal Uni e si y (jsky202205)
Zhang Hongzhi
JiNing No mal Uni e si y, School o Ma hema ics and S a is ic,
Ulanqab, Inne Mongolia, 012000 ,P. R. China
h ps://doi.o g/10.5281/zenodo.17249788
Abs ac
In he knowledge sys em o uni e si y physics, symme y is widely p esen in b anches such as mechanics
and elec omagne ism. I is no only a concep ual ool ha connec s di e en disciplines, bu also demons a es
signi ican alue in sol ing p oblems. This pape conduc s an analysis h ough speci ic examples: In pa icle kin-
ema ics o mechanics, he symme y o mo ion is used o equa e he ajec o y o a small ball’s zigzag elas ic
collisions inside a well o p ojec ile mo ion, simpli ying mul i-p ocess p oblems in o a single model; In elec o-
s a ic ields, aking a uni o mly cha ged hin disk as an example, Gauss's heo em is applied in combina ion wi h
he plana symme y o he elec ic ield o e icien ly sol e o he elec ic ield in ensi y on he axis, a oiding
complex in eg a ions; In s eady magne ic ields, o an in ini ely long cu en -ca ying hin me al pla e, Ampè e's
ci cui al heo em is u ilized, and he mi o symme y o he magne ic ield is employed o simpli y he calcula ion
o magne ic induc ion. The esea ch shows ha he symme y me hod has he common ad an ages o "mo e con-
cise in hinking and analysis, and mo e e icien in calcula ion and solu ion" in sol ing uni e si y physics p ob-
lems. I can help esea che s quickly iden i y p oblem-sol ing pa hs and educe he complexi y o p oblems.
Keywo ds: Symme y; Uni e si y Physics; Gauss's Theo em; Ampè e's Ci cui al Theo em
Symme y e e s o he co esponding o p opo -
ional ela ionship be ween he a ious componen s
wi hin a whole. The o igin o his concep can be aced
back o he ea ly p oduc i e p ac ical ac i i ies o hu-
mans — in he p ocess o obse ing and ans o ming
he wo ld, people i s cap u ed he exis ence o sym-
me y om li e scena ios and na u al phenomena. Fo
example, egula ci cula geome ic pa e ns, lea es
wi h symme ically dis ibu ed lea eins, he symme -
ic le - igh body s uc u es o animals, and he ax-
isymme ic layou s commonly seen in ancien Chinese
a chi ec u e (such as he palace complex o he Fo bid-
den Ci y) a e all in ui i e mani es a ions o symme y.
Wi hin he knowledge sys em o uni e si y phys-
ics, symme y is a co e concep ha uns h ough mul-
iple ields, widely exis ing in subdisciplines such as
mechanics, elec omagne ism, and quan um mechan-
ics. Wi hin he con ex o physics, he de ini ion o
symme y is u he e ined: i a ce ain ope a ion is
pe o med on a physical sys em (i.e., he p ocess o
ansi ioning he sys em om i s ini ial s a e o ano he
s a e), and he co e p ope ies o he sys em a e he
ope a ion a e comple ely equi alen o hose in i s ini-
ial s a e wi hou any changes, hen he sys em is said
o possess symme y wi h espec o his ope a ion.Pu
simply, he essence o symme y can be unde s ood as
he p ope y o a physical sys em ha emains un-
changed unde speci ic changes. Whe he i is spa ial
ope a ions such as ansla ion, o a ion, o e lec ion, o
empo al ope a ions such as ime ansla ion, as long as
he physical laws o he sys em (e.g., equa ions o mo-
ion, ene gy conse a ion ela ionships, e c.) emain
una ec ed a e he ope a ion, he sys em can be de e -
mined o ha e he co esponding symme y.
Symme y occupies a c ucial co e posi ion in he
knowledge sys em o uni e si y physics. I s me hods
and ideas un h ough almos all co e modules such as
mechanics, he modynamics, op ics, and elec omag-
ne ism, se ing as a key concep ual ool o linking di -
e en physics b anches. Among hese, he applica ion
o symme y me hods is pa icula ly p ominen and
widesp ead in he ield o elec omagne ism. Whe he
i is he calcula ion o elec ic ield in ensi y and mag-
ne ic induc ion in ensi y, o he analysis o he laws o
elec omagne ic induc ion, symme y is o en used as
an impo an b eak h ough.
F om he pe spec i e o p ac ical p oblem-sol -
ing, he alue o symme y is e en mo e di ec and no-
able. When sol ing complex physics p oblems, i one
can skill ully u ilize he symme y o he sys em (such
as sphe ical symme y, cylind ical symme y, plana
symme y, e c.), i can g ea ly simpli y he physical
model; a he same ime, i can e ec i ely educe ma h-
ema ical de i a ions and calcula ion s eps, a oid he so-
lu ion o complex in eg als o sys ems o equa ions, and
ul ima ely achie e he goal o ob aining esul s quickly
and simply.
Thanks o i s ea u es o simpli ying models, op i-
mizing solu ions, and e ealing essences, he symme y
me hod has anscended me e p oblem-sol ing ech-
niques. I has become a undamen al hinking and ool
o people o unde s and he s uc u al composi ion o
ma e (such as he symme ic s uc u e o c ys als) and
56 Danish Scien i ic Jou nal No100, 2025
explo e he laws o in e ac ions be ween physical quan-
i ies, p o iding a key pe spec i e o in-dep h comp e-
hension o he o de and laws o he physical wo ld.
To in ui i ely demons a e he con enience o he
symme y me hod in sol ing uni e si y physics p ob-
lems, he ollowing ex will conduc an analysis wi h
speci ic examples. Fi s , we ake a ypical p ojec ile
mo ion p oblem in mechanics as he s a ing poin , o
b ie ly discuss he signi ican ad an ages demons a ed
when using symme y hinking o sol e p oblems.
1. Applica ion o Symme y in Pa icle Kine-
ma ics
Case 1: As shown in Figu e 1, he e is a well wi h
a dep h o H and a smoo h inne wall, and i s inne di-
ame e is d. Now, a small ball is p ojec ed in o he well
om he wellhead along he diame ical di ec ion o he
well. Du ing he alling p ocess, he small ball unde -
goes n elas ic collisions wi h he well wall. When i i-
nally eaches he bo om o he well, i s landing poin
and he ini ial p ojec ion poin lie on he same s aigh
line. T y o ind he ini ial eloci y
0
o he small ball
when i is p ojec ed.
Figu e 1: Schema ic Diag am o a Well wi h Smoo h Inne Walls and a Small Ball's n Elas ic Collisions
I can be known om he p oblem ha he inne
wall o he well is smoo h, and he collision be ween
he small ball and he well wall is an elas ic collision.
A his poin , we can u ilize he symme y o mo ion
and ega d he mo ion o he small ball inside he well
as p ojec ile mo ion. As shown in Figu e 2, he dis ance
a eled by he small ball in he ho izon al di ec ion is
nd. Acco ding o he laws o p ojec ile mo ion: in he
ho izon al di ec ion, he e is
0
nd
; in he e ical
di ec ion, he e is
2
2
1g H
. By combining hese
wo equa ions, we can sol e o
Hg
H
nd
2
2
0
.
Figu e 2: Schema ic Diag am o he Symme y o he Small Ball's Mo ion, Equi alen P ojec ile Mo ion, and So-
lu ion o he Ini ial Veloci y
I can be seen om he abo e example in uni e -
si y physics mechanics ha by skill ully applying he
symme y o mo ion, we equi alen ly ans o m he
b oken-line ebound ajec o y o he small ball inside
he well in o a con inuous p ojec ile mo ion ajec o y.
Based on he cha ac e is ics o symme ic eloci y di-
ec ion and cons an ho izon al speed in elas ic colli-
sions, we achie e an essen ial simpli ica ion o he
complex mo ion p ocess. This p ocessing me hod can
di ec ly s ip away he "b oken-line in e e ence"
caused by collisions, and ans o m he mul i-p ocess
p oblem ha o iginally equi es segmen al analysis in o
a single p ojec ile mo ion model, uly ealizing "sim-
pli ying he complex". I ully demons a es he signi -
ican e ec i eness o he symme y me hod in simpli-
ying physical p ocesses and imp o ing p oblem-sol -
ing e iciency.
2. Applica ion o Symme y in Elec os a ic
Fields
The p e ious sec ion has demons a ed he p ac i-
cal alue o symme y in sol ing uni e si y physics
p oblems h ough he example o p ojec ile mo ion in
mechanics. In his example, we abs ac ed he small
ball as he ideal model o a "pa icle" and u he s ud-
ied i s mo ion laws inside he well — which essen ially
in ol es he explo a ion o he "mechanical mo ion
laws o pa icles" in he ield o mechanics. In he sys-
em o uni e si y physics, howe e , elec omagne ism
is ano he co e b anch; he "elec omagne ic mo ion" i
s udies is ano he undamen al o m o mo ion o ma -
e , dis inc om mechanical mo ion.
Danish Scien i ic Jou nal No100, 2025 57
Uni e si y physics usually analyzes he basic laws
o elec omagne ic mo ion om he pe spec i e o
" ield". Simila o he a ionalized modeling app oach
o "pa icle" in mechanics, elec omagne ism also akes
he "poin cha ge" as he ideal model when s udying
elec os a ic ields, and on his basis deduces he basic
p ope ies and laws o elec os a ic ields. In he de-
sc ip i e sys em o elec os a ic ields, he e a e wo
co e physical quan i ies: elec ic ield in ensi y and
elec ic po en ial. Among hem, elec ic ield in ensi y
is a ec o poin unc ion (i has bo h magni ude and di-
ec ion, and i s alue changes wi h spa ial posi ion), and
elec ic po en ial is a scala poin unc ion (i only has
magni ude, and i s alue also changes wi h spa ial po-
si ion). Fo a cha ged body a es in an ine ial ame
o e e ence, he co e logic o sol ing i s elec os a ic
ield p oblem lies in: i we can calcula e he elec ic
ield in ensi y and elec ic po en ial a each poin in he
elec os a ic ield exci ed by his cha ged body h ough
physical me hods, hen subsequen p oblems ela ed o
his elec os a ic ield can be easily sol ed.
Case 2: As shown in Figu e 3, conside a uni-
o mly cha ged hin disk wi h a su ace cha ge densi y
o σ (whe e σ is a cons an ) and a adius o R. We now
need o ind he elec ic ield in ensi y a any poin on
he axis ha passes h ough he cen e o he disk and is
pe pendicula o i s su ace.
Figu e 3: Schema ic Diag am o he Solu ion o he Elec ic Field In ensi y on he Axis o a Uni o mly
Cha ged Thin Disk
Gauss's Law is one o he impo an me hods o
sol ing elec ic ield in ensi y in elec omagne ism.
The ma hema ical exp ession o Gauss's Law o elec-
os a ic ields in acuum is:
0
1
sE dS q
.
The physical meaning o his exp ession can be clea ly
s a ed as: In acuum, he elec ic lux o he elec ic
ield in ensi y h ough any closed su ace (known as he
Gaussian su ace) is equal o he a io o he algeb aic
sum o he cha ges enclosed by his Gaussian su ace o
he pe mi i i y o ee space.I should be pa icula ly
emphasized ha he e icien applica ion o Gauss's
Law depends on he symme y o he elec ic ield —
only when he elec ic ield exci ed by a cha ged body
has a egula ly symme ic dis ibu ion (such as sphe i-
cal symme y, cylind ical symme y, o plana sym-
me y) can a sui able Gaussian su ace be selec ed. This
selec ion ensu es ha he magni ude o he elec ic ield
in ensi y is he same e e ywhe e on he Gaussian su -
ace (o ze o in some egions), and he di ec ion o he
elec ic ield in ensi y is consis en wi h (o pe pendic-
ula o) he di ec ion o he a ea elemen ec o . In his
way, he in eg al calcula ion o he elec ic ield in en-
si y lux is simpli ied. The e o e, Gauss's Law can be
ega ded as he co e me hod o sol ing he elec ic
ield in ensi y using symme y in elec os a ic ields.
The gene al s eps o sol ing he elec ic ield in ensi y
using Gauss's Law a e as ollows:
(1) Analyze he symme y o he elec ic ield:
Fi s , judge he cha ge dis ibu ion cha ac e is ics o he
cha ged body (such as a uni o mly cha ged sphe ical
su ace, an in ini ely long uni o mly cha ged cylind i-
cal su ace, and so on), hen de e mine he symme y
ype o he elec ic ield i exci es (sphe ical symme y,
cylind ical symme y, e c.), and cla i y he di ec ional
egula i y o he elec ic ield in ensi y and he cha ac-
e is ics o i s magni ude dis ibu ion;
(2) Selec an app op ia e Gaussian su ace: Based
on he symme y o he elec ic ield, selec a Gaussian
su ace ha ma ches he elec ic ield dis ibu ion (e.g.,
a sphe ical Gaussian su ace o a sphe ically symme -
ic elec ic ield, and a coaxial cylind ical Gaussian su -
ace o a cylind ically symme ic elec ic ield), and
ensu e ha he magni ude o he elec ic ield in ensi y
is he same e e ywhe e on he Gaussian su ace (o ze o
in a ce ain egion), and ha he angle be-
ween
E
and
dS
(a ea elemen ec o ) is cons an
(usually 0° o 90°, which acili a es in eg al calcula-
ion);
(3) Calcula e he elec ic ield in ensi y lux and
he enclosed cha ge: On one hand, calcula e he elec ic
ield in ensi y lux h ough he Gaussian su ace; on he
o he hand, based on he cha ge dis ibu ion o he
cha ged body, calcula e he algeb aic sum o he en-
closed cha ge inside he Gaussian su ace;
(4) Apply Gauss's Law o sol e o he elec ic
ield in ensi y: Subs i u e he wo calcula ion esul s
men ioned abo e in o he o mula o Gauss's Law, and
a e ea angemen , you can sol e o he magni ude o
he elec ic ield in ensi y. Then, combine i wi h he
p e iously analyzed di ec ional egula i y o he elec-
ic ield o inally de e mine he elec ic ield in en-
si y
E
;
58 Danish Scien i ic Jou nal No100, 2025
When he dis ance x om he ield poin P o he
cen e O o he hin disk is much smalle
han R (whe e R is he adius o he disk), since he
ield poin is ex emely close o he disk, he edge e ec
o he disk can be neglec ed. A his ime, he uni o mly
cha ged hin disk can be app oxima ed as an "in ini ely
la ge uni o mly cha ged plane". Fo an in ini ely la ge
uni o mly cha ged plane, he elec ic ield i exci es has
plane symme y: he di ec ion o he elec ic ield in-
ensi y on bo h sides o he plane is pe pendicula o he
plane ( o a posi i ely cha ged plane, he elec ic ield
di ec ion is away om he plane; o a nega i ely
cha ged plane, i poin s owa ds he plane), and in he
same plane pa allel o he cha ged plane, he magni ude
o he elec ic ield in ensi y a each poin is equal e e-
ywhe e. Based on his symme y ea u e, his p oblem
can be sol ed e icien ly by he symme y me hod —
Gauss's Law.
As shown in Figu e 4, based on he plane sym-
me y o he elec ic ield, a cylind ical su ace whose
axis is pe pendicula o he cha ged plane is selec ed as
he Gaussian su ace: he wo ci cula bases o he cyl-
inde a e pa allel o he cha ged plane and symme ic
wi h espec o he plane ( o ensu e ha he magni ude
o he elec ic ield in ensi y is equal a he wo bases),
and he la e al su ace is pe pendicula o he cha ged
plane.
Since he no mal o he la e al su ace o he cyl-
inde is pe pendicula o he elec ic ield in ensi y,
he elec ic ield in ensi y lux h ough he la e al su -
ace is ze o. Fu he mo e, because he no mals o he
cylinde ’s ci cula bases a e pa allel o he elec ic ield
in ensi y, and he magni ude o he elec ic ield in en-
si y is he same a bo h bases, he o al elec ic ield in-
ensi y lux h ough he wo ci cula bases is
2ES (whe e S is he a ea o a single ci cula base). In
addi ion, gi en ha he su ace cha ge densi y o he
cha ged plane is σ, acco ding o Gauss's Law, we ha e
0
2S
ES
, ha is,
0
2
E
.
Figu e 4: Schema ic Diag am o he Selec ion o a Cylind ical Gaussian Su ace Based on he Plana Symme y
o he Elec ic Field
3. Applica ion o Symme y in S eady Magne ic
Fields
F om he physical examples in he p e ious ex ,
we know ha elec os a ic ields a e induced a ound
s a iona y cha ges, and he analysis o such elec ic
ields can be simpli ied using symme y (e.g., Gauss's
Law). Fu he esea ch shows ha when cha ges mo e
o o m a cu en , he ield dis ibu ion a ound hem
changes — a his ime, no only does an elec ic ield
exis , bu a s eady magne ic ield is also induced (i he
magni ude and di ec ion o he cu en do no change
wi h ime, he co esponding magne ic ield is a s eady
magne ic ield).
This phenomenon e eals he deep connec ion be-
ween elec ici y and magne ism: he wo a e no iso-
la ed bu o m a uni ied whole ha is in e connec ed
and mu ually ans o mable ( o example, he phenom-
enon o elec omagne ic induc ion demons a es mag-
ne ism gene a ing elec ici y, while he magne ic e ec
o cu en demons a es elec ici y gene a ing mag-
ne ism). Fo his eason, in p ac ical applica ions, sce-
na ios in ol ing elec ici y (such as ci cui s and elec i-
cal equipmen ) a e o en accompanied by he in ol e-
men o magne ism, and he analysis o s eady magne ic
ields has hus become an impo an pa o he elec o-
magne ism module in uni e si y physics.
Case 3: As shown in Figu e 5, he e is an in ini ely
long hin me al pla e wi h a wid h o 2d, h ough which
a s eady cu en I lows uni o mly ( he cu en di ec ion
is along he leng h o he me al pla e). I is equi ed o
sol e o he magne ic induc ion in ensi y
B
a a a ge
ield poin P, whe e P lies on a plane ha passes
h ough he midline o he me al pla e (i.e., he cen al
line pe pendicula o he wid h di ec ion o he pla e)
and is pe pendicula o he su ace o he me al pla e.
Danish Scien i ic Jou nal No100, 2025 59
Figu e 5: Schema ic Diag am o he In ini ely Long Cu en -Ca ying Thin Me al Pla e and he Posi ion o Ta -
ge Field Poin P
In he analysis o s eady magne ic ields, i he
magne ic ield dis ibu ion exhibi s egula symme y,
he magne ic induc ion in ensi y can be e icien ly
sol ed using Ampè e's Ci cui al Law — a app oach
highly simila o ha o using Gauss's Law o sol e o
elec ic ield in ensi y in elec os a ic ields. The ma h-
ema ical exp ession o Ampè e's Ci cui al Law
is:
0
LB dl I
(whe e L is any closed
pa h,
B
is he magne ic induc ion in ensi y a each
poin on he pa h,
dl
is he line elemen ec o o he
pa h,
0
is he pe meabili y o ee space,
and
I
is he algeb aic sum o he cu en s enclosed
by he closed pa h L; he cu en di ec ion is posi i e i
i sa is ies he igh -hand sc ew ule wi h he di ec ion
o he loop’s a e sal, and nega i e o he wise).
The physical meaning o his heo em can be
clea ly s a ed as ollows: In a s eady magne ic ield in
acuum, he line in eg al o he magne ic induc ion in-
ensi y
B
along any closed pa h (i.e., he ci cula ion)
is equal o he p oduc o he pe meabili y o ee space
and he algeb aic sum o all cu en s enclosed by he
closed pa h.I can be seen om his ha he co e p e-
equisi e o sol ing he magne ic induc ion in ensi y
using Ampè e's Ci cui al Law is he symme y o he
magne ic ield. Only in his way can an app op ia e
closed pa h (i.e., an Ampè e loop) be selec ed—such
ha he magni ude o he magne ic induc ion in ensi y
is uni o m a all poin s on he loop (o ze o in some sec-
ions), and i s di ec ion is consis en wi h (o pe pendic-
ula o) he di ec ion o he loop’s line elemen . This
simpli ies he in eg al calcula ion o he ci cula ion.
Re u ning o Case 3, h ough heo e ical analysis,
we ans o m he a o emen ioned in ini ely long cu -
en -ca ying hin me al pla e in o a la s uc u e com-
posed o a numbe o mu ually pa allel in ini ely long
cu en -ca ying s aigh wi es. A his poin , he mag-
ne ic ield induced by he cu en -ca ying me al pla e
exhibi s symme y.
Nex , we use Ampè e's Ci cui al Law o calcula e
he magne ic induc ion in ensi y
B
a any poin ou -
side he cu en -ca ying plane. Since he magne ic
ield induced by he plane exhibi s symme y, he mag-
ni ude o he magne ic induc ion in ensi y
B
is equal
a all poin s equidis an om he plane. Addi ionally,
he di ec ion o
B
on ei he side o he plane is pa allel
o he plane, while he di ec ions o
B
on he wo sides
a e opposi e o each o he .
As shown in Figu e 6, selec a closed pa h abcda,
whe e segmen s ab and cd a e bo h pa allel o
B
, and
segmen bc is pe pendicula o
B
. Acco ding o Am-
pè e's Ci cui al Law, we ha e:
00
b c d a
a b c d
L
B dl B dl B dl B dl B dl I I ab
Since
0
B ab B cd i ab
,and
ab cd
， he e o e
0
2B ab i ab
， ha is
0
2
Bi
.
60 Danish Scien i ic Jou nal No100, 2025
Figu e 6: Schema ic Diag am o he Ampè e Loop o he Solu ion o Magne ic Induc ion In ensi y o a
Cu en -Ca ying Plane
F om he analysis o he abo e examples, a gene al
conclusion can be d awn: When he symme y me hod
is applied o sol ing uni e si y physics p oblems, e-
ga dless o which b anch o physics he p oblem in-
ol es, i s co e ad an ages a e e lec ed in wo aspec s
— namely, mo e concise hinking and analysis, and
mo e e icien calcula ion and solu ion. P ecisely be-
cause o his, in sol ing p oblems and conduc ing e-
sea ch in uni e si y physics, people o en p io i ize
judging whe he he physical scena io exhibi s sym-
me y. I symme ic cha ac e is ics exis , a simple so-
lu ion pa h can be quickly iden i ied, signi ican ly e-
ducing he complexi y o he p oblem. Fu he mo e, he
symme y me hod is no only a "sho cu " o p oblem-
sol ing; i also plays an i eplaceable and posi i e d i -
ing ole in explo ing physical models (e.g., equa ing
complex mo ions o ideal models), unco e ing physical
mys e ies (e.g., e ealing he laws o in e ac ion be-
ween ields and ma e ), and pe cei ing he inhe en
ha monious beau y o physics (e.g., he uni y o laws
b ough abou by symme ic dis ibu ion).
Re e ences:
1. Ma, W. W., Xie, X. S., & Zhou, Y. Q. Phys-
ics (5 h Edi ion). Beijing: Highe Educa ion P ess,
2006.
2. Ma, W. W., Chen, G. Q., & Chen, J. S udy
Guide o Physics (5 h Edi ion). Beijing: Highe
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