Coordinate System & Vector Analysis Short notes | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download

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Short Notes on Co ordinate System and V ector Analysis
1. In tro duction
• Co ordinate systems and v ector analysis are essen tial to ols in electromagnetics for describing electric
and magnetic fields.
• Co ordinate systems define p ositions in space, while v ector analysis manipulates field quan tities
(e.g.,E ,B ).
• Common applications: Solving Maxw ell’s equations, analyzing field b eha vior.
2. Co ordinate Systems
• Cartesian (Rectangular) : (x,y,z) , unit v ectors ˆ x,ˆ y,ˆ z .
– V ector: A = A
x
ˆ x+A
y
ˆ y +A
z
ˆ z .
– Differen tial length: dl = dxˆ x+dyˆ y +dzˆ z .
• Cylindrical : (?,?,z) , unit v ectors ˆ ?,
ˆ
?,ˆ z .
– V ector: A = A
?
ˆ ?+A
?
ˆ
?+A
z
ˆ z .
– Differen tial length: dl = d?ˆ ?+?d?
ˆ
?+dzˆ z .
• Spherical : (r,?,?) , unit v ectors ˆ r,
ˆ
?,
ˆ
? .
– V ector: A = A
r
ˆ r +A
?
ˆ
? +A
?
ˆ
? .
– Differen tial length: dl = drˆ r +rd?
ˆ
? +rsin?d?
ˆ
? .
• Choice dep ends on problem symmetry (e.g., cylindrical for wires, spherical for p oin t c harges).
3. V ector Op erations
• A ddition/Subtraction : A±B = (A
x
±B
x
)ˆ x+(A
y
±B
y
)ˆ y +(A
z
±B
z
)ˆ z .
• Scalar Multiplication : kA = kA
x
ˆ x+kA
y
ˆ y +kA
z
ˆ z .
• Dot Pro duct : A·B = ABcos? = A
x
B
x
+A
y
B
y
+A
z
B
z
.
• Cross Pro duct : A×B =







ˆ x ˆ y ˆ z
A
x
A
y
A
z
B
x
B
y
B
z







.
• Magnitude : |A| =
q
A
2
x
+A
2
y
+A
2
z
.
4. Differen tial V ector Op erators
• Gradien t : F or scalar field V , ?V represen ts the direction of maxim um increase.
Cartesian: ?V =
?V
?x
ˆ x+
?V
?y
ˆ y +
?V
?z
ˆ z
• Div ergence : Measures the “outflo w” of v ector field A .
Cartesian: ?·A =
?A
x
?x
+
?A
y
?y
+
?A
z
?z
1
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Short Notes on Co ordinate System and V ector Analysis
1. In tro duction
• Co ordinate systems and v ector analysis are essen tial to ols in electromagnetics for describing electric
and magnetic fields.
• Co ordinate systems define p ositions in space, while v ector analysis manipulates field quan tities
(e.g.,E ,B ).
• Common applications: Solving Maxw ell’s equations, analyzing field b eha vior.
2. Co ordinate Systems
• Cartesian (Rectangular) : (x,y,z) , unit v ectors ˆ x,ˆ y,ˆ z .
– V ector: A = A
x
ˆ x+A
y
ˆ y +A
z
ˆ z .
– Differen tial length: dl = dxˆ x+dyˆ y +dzˆ z .
• Cylindrical : (?,?,z) , unit v ectors ˆ ?,
ˆ
?,ˆ z .
– V ector: A = A
?
ˆ ?+A
?
ˆ
?+A
z
ˆ z .
– Differen tial length: dl = d?ˆ ?+?d?
ˆ
?+dzˆ z .
• Spherical : (r,?,?) , unit v ectors ˆ r,
ˆ
?,
ˆ
? .
– V ector: A = A
r
ˆ r +A
?
ˆ
? +A
?
ˆ
? .
– Differen tial length: dl = drˆ r +rd?
ˆ
? +rsin?d?
ˆ
? .
• Choice dep ends on problem symmetry (e.g., cylindrical for wires, spherical for p oin t c harges).
3. V ector Op erations
• A ddition/Subtraction : A±B = (A
x
±B
x
)ˆ x+(A
y
±B
y
)ˆ y +(A
z
±B
z
)ˆ z .
• Scalar Multiplication : kA = kA
x
ˆ x+kA
y
ˆ y +kA
z
ˆ z .
• Dot Pro duct : A·B = ABcos? = A
x
B
x
+A
y
B
y
+A
z
B
z
.
• Cross Pro duct : A×B =







ˆ x ˆ y ˆ z
A
x
A
y
A
z
B
x
B
y
B
z







.
• Magnitude : |A| =
q
A
2
x
+A
2
y
+A
2
z
.
4. Differen tial V ector Op erators
• Gradien t : F or scalar field V , ?V represen ts the direction of maxim um increase.
Cartesian: ?V =
?V
?x
ˆ x+
?V
?y
ˆ y +
?V
?z
ˆ z
• Div ergence : Measures the “outflo w” of v ector field A .
Cartesian: ?·A =
?A
x
?x
+
?A
y
?y
+
?A
z
?z
1
• Curl : Measures the “rotation” of v ector field A .
Cartesian: ?×A =







ˆ x ˆ y ˆ z
?
?x
?
?y
?
?z
A
x
A
y
A
z







• Expressions in cylindrical and spherical co ordinates are more complex, in v olving scale factors.
5. In tegral Theorems
• Div ergence Theorem :
RR
V
?·AdV =
H
S
A·dS .
• Stok es’ Theorem :
RR
S
(?×A)·dS =
H
C
A·dl .
• Used to relate field prop erties in electromagnetics (e.g., Gauss’s la w, Amp ere’s la w).
6. Applications in Electromagnetics
• Electric Field : E =-?V , where V is the ele ctric p oten tial.
• Magnetic Field : ?×B = µ
0
J (Amp ere’s la w in steady state).
• Charge Densit y : ?·D = ? (Gauss’s la w).
• Co ordinate system c hoice simplifies field calculations (e.g., spherical for p oin t c harges).
7. Practical Considerations
• Co ordinate Con v ersion : T ransformations b et w een Cartesian, cylindrical, and spherical systems
are needed for complex geometries.
• Symmetry Exploitation : Simplifies v ector op erations (e.g., radial symmetry in spherical co or-
dinates).
• Numerical Precision : Required for computations i n non-Cartesian systems.
8. Conclusion
• Co ordinate systems and v ector analysis pro vide a framew ork for analyzing electromagnetic fields.
• Prop er c hoice of co ordinate system and mastery of v ector op erations (gradien t, div ergence, curl)
are crucial for solving Maxw ell’s equations e?icien tly .
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