Short Notes: Electrical & Magnetic Fields

 Page 2


 
 
 
 
 
 
Electric field intensity  
 
R
2
FQ
ˆ Ea
q
4R
??
??
 
Electric field direction is away from a positive charge & towards negative charge. 
Charge densities 
1) Linear charge density 
It is denoted by '' ? . It is equal to charge per unit length. 
  ? ?
q
cm
l
?? 
 
2) Surface charge density 
It is denoted by '' ? . It is equal to charge per unit area. 
  
? ?
??
2
q
cm
A
 
 
3) Volume charge density 
It is denoted by '' ? . It is equal to charge per unit volume. 
  
? ?
??
3
q
cm
V
 
Electric field due to continuous charge distribution 
1) Infinite line charge 
Electric field intensity at a distance ‘r’ from a line charge of linear charge density ? 
  
?
?
??
r
o
ˆ Ea
2r
 
 
2) Infinite sheet charge 
Electric field at a distance ‘h’ from an infinite charged sheet with charge density ? is 
  
?
??
?
nn
ˆˆ E a ; a
2
 Normal unit vector  
 
3) Conducting sphere 
If a conducting sphere of radius ‘R’ is charged with a charge ‘Q’ then electric field. 
  
? ?
?
?
?
?
?
?? ?
2
0                 r R
E 
Q
       r R
4r
 
Electric field inside conducting sphere is zero. 
Page 3


 
 
 
 
 
 
Electric field intensity  
 
R
2
FQ
ˆ Ea
q
4R
??
??
 
Electric field direction is away from a positive charge & towards negative charge. 
Charge densities 
1) Linear charge density 
It is denoted by '' ? . It is equal to charge per unit length. 
  ? ?
q
cm
l
?? 
 
2) Surface charge density 
It is denoted by '' ? . It is equal to charge per unit area. 
  
? ?
??
2
q
cm
A
 
 
3) Volume charge density 
It is denoted by '' ? . It is equal to charge per unit volume. 
  
? ?
??
3
q
cm
V
 
Electric field due to continuous charge distribution 
1) Infinite line charge 
Electric field intensity at a distance ‘r’ from a line charge of linear charge density ? 
  
?
?
??
r
o
ˆ Ea
2r
 
 
2) Infinite sheet charge 
Electric field at a distance ‘h’ from an infinite charged sheet with charge density ? is 
  
?
??
?
nn
ˆˆ E a ; a
2
 Normal unit vector  
 
3) Conducting sphere 
If a conducting sphere of radius ‘R’ is charged with a charge ‘Q’ then electric field. 
  
? ?
?
?
?
?
?
?? ?
2
0                 r R
E 
Q
       r R
4r
 
Electric field inside conducting sphere is zero. 
 
 
 
 
 
Electrical potential 
The amount of work done in bringing a unit positive charge from infinity to a certain point in an 
electric field is called electric potential. 
  
?
??
?
A
A
V E.dL 
  ? ? ? EV 
  ? = represent gradiant 
For vector operations, refer engineering mathematics k-notes. 
Electric Flux Density 
  DE ?? 
Electrical flux 
S
D.dS ??
?
 
SI unit of electric flux is coulomb. 
Gauss’s law 
It states that total electric flux through any closed surface is equal to charge enclosed by that 
surface. 
  ??
??
Sb
D.dS dV 
  By Gauss’s Divergence theorem 
  ? ? ? .D 
Magnetic flux Density 
Magnetic flux per unit area is called magnetic flux density. It is a vector quantity and denoted by 
B & its unit is tesla (T). 
Flux B. dS ??
?
 
 
 
Page 4


 
 
 
 
 
 
Electric field intensity  
 
R
2
FQ
ˆ Ea
q
4R
??
??
 
Electric field direction is away from a positive charge & towards negative charge. 
Charge densities 
1) Linear charge density 
It is denoted by '' ? . It is equal to charge per unit length. 
  ? ?
q
cm
l
?? 
 
2) Surface charge density 
It is denoted by '' ? . It is equal to charge per unit area. 
  
? ?
??
2
q
cm
A
 
 
3) Volume charge density 
It is denoted by '' ? . It is equal to charge per unit volume. 
  
? ?
??
3
q
cm
V
 
Electric field due to continuous charge distribution 
1) Infinite line charge 
Electric field intensity at a distance ‘r’ from a line charge of linear charge density ? 
  
?
?
??
r
o
ˆ Ea
2r
 
 
2) Infinite sheet charge 
Electric field at a distance ‘h’ from an infinite charged sheet with charge density ? is 
  
?
??
?
nn
ˆˆ E a ; a
2
 Normal unit vector  
 
3) Conducting sphere 
If a conducting sphere of radius ‘R’ is charged with a charge ‘Q’ then electric field. 
  
? ?
?
?
?
?
?
?? ?
2
0                 r R
E 
Q
       r R
4r
 
Electric field inside conducting sphere is zero. 
 
 
 
 
 
Electrical potential 
The amount of work done in bringing a unit positive charge from infinity to a certain point in an 
electric field is called electric potential. 
  
?
??
?
A
A
V E.dL 
  ? ? ? EV 
  ? = represent gradiant 
For vector operations, refer engineering mathematics k-notes. 
Electric Flux Density 
  DE ?? 
Electrical flux 
S
D.dS ??
?
 
SI unit of electric flux is coulomb. 
Gauss’s law 
It states that total electric flux through any closed surface is equal to charge enclosed by that 
surface. 
  ??
??
Sb
D.dS dV 
  By Gauss’s Divergence theorem 
  ? ? ? .D 
Magnetic flux Density 
Magnetic flux per unit area is called magnetic flux density. It is a vector quantity and denoted by 
B & its unit is tesla (T). 
Flux B. dS ??
?
 
 
 
 
 
 
 
 
Magnetic field intensity 
Represented by H. 
  BH ?? 
  ? = permeability. 
  ? ? ? ?
or
 
  ?
r
 = relative permeability  
  ?
o
 = permeability of free space 
  
?
? ? ? ?
7
o
4 10 H m
 
 
Biot – Savart’s law 
  
? ?
??
?
R 2
I
ˆ d H dL a
4R
 
Magnetic field due to infinite line current 
  
I
ˆ Ha
2
?
?
??
 
 ? = perpendicular distance of point from line current. 
 
ˆ a
?
 = Unit vector in cylindrical co-ordinates. 
 
 
Ampere’s Circuital law 
It states that line integral of magnetic field intensity H around any closed path is exactly equal to 
net current enclosed by that path. 
  
enclosed
H . dL I ?
?
 
  H. dL J . ds ?
??
 
 By stokes theorem 
  ? ? ? HJ 
 
 
Page 5


 
 
 
 
 
 
Electric field intensity  
 
R
2
FQ
ˆ Ea
q
4R
??
??
 
Electric field direction is away from a positive charge & towards negative charge. 
Charge densities 
1) Linear charge density 
It is denoted by '' ? . It is equal to charge per unit length. 
  ? ?
q
cm
l
?? 
 
2) Surface charge density 
It is denoted by '' ? . It is equal to charge per unit area. 
  
? ?
??
2
q
cm
A
 
 
3) Volume charge density 
It is denoted by '' ? . It is equal to charge per unit volume. 
  
? ?
??
3
q
cm
V
 
Electric field due to continuous charge distribution 
1) Infinite line charge 
Electric field intensity at a distance ‘r’ from a line charge of linear charge density ? 
  
?
?
??
r
o
ˆ Ea
2r
 
 
2) Infinite sheet charge 
Electric field at a distance ‘h’ from an infinite charged sheet with charge density ? is 
  
?
??
?
nn
ˆˆ E a ; a
2
 Normal unit vector  
 
3) Conducting sphere 
If a conducting sphere of radius ‘R’ is charged with a charge ‘Q’ then electric field. 
  
? ?
?
?
?
?
?
?? ?
2
0                 r R
E 
Q
       r R
4r
 
Electric field inside conducting sphere is zero. 
 
 
 
 
 
Electrical potential 
The amount of work done in bringing a unit positive charge from infinity to a certain point in an 
electric field is called electric potential. 
  
?
??
?
A
A
V E.dL 
  ? ? ? EV 
  ? = represent gradiant 
For vector operations, refer engineering mathematics k-notes. 
Electric Flux Density 
  DE ?? 
Electrical flux 
S
D.dS ??
?
 
SI unit of electric flux is coulomb. 
Gauss’s law 
It states that total electric flux through any closed surface is equal to charge enclosed by that 
surface. 
  ??
??
Sb
D.dS dV 
  By Gauss’s Divergence theorem 
  ? ? ? .D 
Magnetic flux Density 
Magnetic flux per unit area is called magnetic flux density. It is a vector quantity and denoted by 
B & its unit is tesla (T). 
Flux B. dS ??
?
 
 
 
 
 
 
 
 
Magnetic field intensity 
Represented by H. 
  BH ?? 
  ? = permeability. 
  ? ? ? ?
or
 
  ?
r
 = relative permeability  
  ?
o
 = permeability of free space 
  
?
? ? ? ?
7
o
4 10 H m
 
 
Biot – Savart’s law 
  
? ?
??
?
R 2
I
ˆ d H dL a
4R
 
Magnetic field due to infinite line current 
  
I
ˆ Ha
2
?
?
??
 
 ? = perpendicular distance of point from line current. 
 
ˆ a
?
 = Unit vector in cylindrical co-ordinates. 
 
 
Ampere’s Circuital law 
It states that line integral of magnetic field intensity H around any closed path is exactly equal to 
net current enclosed by that path. 
  
enclosed
H . dL I ?
?
 
  H. dL J . ds ?
??
 
 By stokes theorem 
  ? ? ? HJ 
 
 
 
 
 
 
 
Maxwell equations 
1) 
dB
E . dL B . dS or E
dt t
? ? ?
? ? ? ?
?
? ? ?
 
2) 
1
E . dS dv or . E
?
? ? ? ?
?
?
? ? ? ? ?
 
3) B . dS 0 or . B 0 ? ? ?
??
 
4) ? ? ? ? ?
? ? ? ? ? 0 o o
d
B . dL J. ds E . ds
dt
 
                        or 
??
??
??
??
?
? ? ? ? ? ?
?
oo
E
B J 
t