Electromagnetic Induction | Physics Optional Notes for UPSC PDF Download

 Page 1


ELECTROMAGNETIC INDUCTION (EMI) 
INTRODUCTION 
 
In 1820, Oersted found out that electric current makes magnets move. Later, Faraday 
had an idea: what if it works the other way too? What if magnets can make electric 
current? Faraday tested this by moving a magnet near a coil of wire connected to a 
sensitive device called a galvanometer. When the magnet moved through the coil, the 
galvanometer immediately showed a change.  
Magnetic Flux (?? ) : 
Magnetic flux through an area ?? is given by ?? 
?? =???? 
·?? ?? 
 ??? 
= magnetic field intensity 
 
SI unit of ?? is Weber (Wb) or Tesla m
2
. 
If B
?? 
 is constant 
?? =B
?? 
·A
?? 
 A
?? 
= Area vector whose direction is normal to the plane of area. 
?? =|B
?? 
|·|A
?? 
|cos ?? 
Note: Any one the two normal can be taken as direction of area vector but throughout a 
single calculation conversation taken should not be charged. 
FLUX LINKAGE 
If a coil has more than one turn, then the flux through the whole coil is the sum of the 
fluxes through the individual turns. If the magnetic field is uniform, the flux through one 
turn is ?? =BAcos ?? If the coil has N turns, the total flux linkage ?? = NBA cos ?? 
? Magnetic lines of force are imaginary, magnetic flux is a real scalar physical 
quantity with dimensions [?? ]=B× area =[
F
IL
][L
2
] 
? B=
F
ILsin ?? ? [?? ]=[
???? ?? -2
AL
][L
2
]=[?? ?? 2
 T
-2
 A
-1
]
 
Page 2


ELECTROMAGNETIC INDUCTION (EMI) 
INTRODUCTION 
 
In 1820, Oersted found out that electric current makes magnets move. Later, Faraday 
had an idea: what if it works the other way too? What if magnets can make electric 
current? Faraday tested this by moving a magnet near a coil of wire connected to a 
sensitive device called a galvanometer. When the magnet moved through the coil, the 
galvanometer immediately showed a change.  
Magnetic Flux (?? ) : 
Magnetic flux through an area ?? is given by ?? 
?? =???? 
·?? ?? 
 ??? 
= magnetic field intensity 
 
SI unit of ?? is Weber (Wb) or Tesla m
2
. 
If B
?? 
 is constant 
?? =B
?? 
·A
?? 
 A
?? 
= Area vector whose direction is normal to the plane of area. 
?? =|B
?? 
|·|A
?? 
|cos ?? 
Note: Any one the two normal can be taken as direction of area vector but throughout a 
single calculation conversation taken should not be charged. 
FLUX LINKAGE 
If a coil has more than one turn, then the flux through the whole coil is the sum of the 
fluxes through the individual turns. If the magnetic field is uniform, the flux through one 
turn is ?? =BAcos ?? If the coil has N turns, the total flux linkage ?? = NBA cos ?? 
? Magnetic lines of force are imaginary, magnetic flux is a real scalar physical 
quantity with dimensions [?? ]=B× area =[
F
IL
][L
2
] 
? B=
F
ILsin ?? ? [?? ]=[
???? ?? -2
AL
][L
2
]=[?? ?? 2
 T
-2
 A
-1
]
 
SI UNIT of magnetic flux: 
? [ML
2
 T
-2
] Corresponds to energy 
 joule 
 ampere 
=
 joule × second 
 coulomb 
= weber (Wb) 
Or  T-m
2
( as tesla =Wb/m
2
) [ ampere =
 coulomb 
 sec ond 
] 
? For a given area flux will be maximum: 
 
When magnetic field ??? 
 is normal to the area 
?? =0
°
 ?cos ?? =maximum=1 ?? max
=BA 
For a given area flux will be minimum: 
When magnetic field ??? 
 is parallel to the area 
 
?? =90
°
?cos ?? = minimum =0 ?? min
=0 
Example: Find flux associated with given areas. 
1. 
 
3. 
 
Page 3


ELECTROMAGNETIC INDUCTION (EMI) 
INTRODUCTION 
 
In 1820, Oersted found out that electric current makes magnets move. Later, Faraday 
had an idea: what if it works the other way too? What if magnets can make electric 
current? Faraday tested this by moving a magnet near a coil of wire connected to a 
sensitive device called a galvanometer. When the magnet moved through the coil, the 
galvanometer immediately showed a change.  
Magnetic Flux (?? ) : 
Magnetic flux through an area ?? is given by ?? 
?? =???? 
·?? ?? 
 ??? 
= magnetic field intensity 
 
SI unit of ?? is Weber (Wb) or Tesla m
2
. 
If B
?? 
 is constant 
?? =B
?? 
·A
?? 
 A
?? 
= Area vector whose direction is normal to the plane of area. 
?? =|B
?? 
|·|A
?? 
|cos ?? 
Note: Any one the two normal can be taken as direction of area vector but throughout a 
single calculation conversation taken should not be charged. 
FLUX LINKAGE 
If a coil has more than one turn, then the flux through the whole coil is the sum of the 
fluxes through the individual turns. If the magnetic field is uniform, the flux through one 
turn is ?? =BAcos ?? If the coil has N turns, the total flux linkage ?? = NBA cos ?? 
? Magnetic lines of force are imaginary, magnetic flux is a real scalar physical 
quantity with dimensions [?? ]=B× area =[
F
IL
][L
2
] 
? B=
F
ILsin ?? ? [?? ]=[
???? ?? -2
AL
][L
2
]=[?? ?? 2
 T
-2
 A
-1
]
 
SI UNIT of magnetic flux: 
? [ML
2
 T
-2
] Corresponds to energy 
 joule 
 ampere 
=
 joule × second 
 coulomb 
= weber (Wb) 
Or  T-m
2
( as tesla =Wb/m
2
) [ ampere =
 coulomb 
 sec ond 
] 
? For a given area flux will be maximum: 
 
When magnetic field ??? 
 is normal to the area 
?? =0
°
 ?cos ?? =maximum=1 ?? max
=BA 
For a given area flux will be minimum: 
When magnetic field ??? 
 is parallel to the area 
 
?? =90
°
?cos ?? = minimum =0 ?? min
=0 
Example: Find flux associated with given areas. 
1. 
 
3. 
 
4. 
 
5. 
 
??ˆ
? 
?? =?? ?? ?? 2
?? =0
 
??ˆ
=? 
?? =-B
(Lx)
2
 
?? =BAcos 60
°
=
BA
2
 
?? =BAcos 120
°
=-
BA
2
 
6. 
 
B
0
?? i
=-BA
?? i
=BA
??? =2BA
A
ˆ
=?
|?? b
|=|?? f
-?? i
 
Example. A loop of area 0.06 m
2
 is placed in a magnetic field 1.2 T with its plane 
inclined 30
°
 to the field direction. Find the flux linked with plane of loop. 
Solution: Area of loop is 0.06 m
2
, B=1.2 T and ?? =90
°
-30
°
=60
°
 
So, the flux linked with the loop is 
Page 4


ELECTROMAGNETIC INDUCTION (EMI) 
INTRODUCTION 
 
In 1820, Oersted found out that electric current makes magnets move. Later, Faraday 
had an idea: what if it works the other way too? What if magnets can make electric 
current? Faraday tested this by moving a magnet near a coil of wire connected to a 
sensitive device called a galvanometer. When the magnet moved through the coil, the 
galvanometer immediately showed a change.  
Magnetic Flux (?? ) : 
Magnetic flux through an area ?? is given by ?? 
?? =???? 
·?? ?? 
 ??? 
= magnetic field intensity 
 
SI unit of ?? is Weber (Wb) or Tesla m
2
. 
If B
?? 
 is constant 
?? =B
?? 
·A
?? 
 A
?? 
= Area vector whose direction is normal to the plane of area. 
?? =|B
?? 
|·|A
?? 
|cos ?? 
Note: Any one the two normal can be taken as direction of area vector but throughout a 
single calculation conversation taken should not be charged. 
FLUX LINKAGE 
If a coil has more than one turn, then the flux through the whole coil is the sum of the 
fluxes through the individual turns. If the magnetic field is uniform, the flux through one 
turn is ?? =BAcos ?? If the coil has N turns, the total flux linkage ?? = NBA cos ?? 
? Magnetic lines of force are imaginary, magnetic flux is a real scalar physical 
quantity with dimensions [?? ]=B× area =[
F
IL
][L
2
] 
? B=
F
ILsin ?? ? [?? ]=[
???? ?? -2
AL
][L
2
]=[?? ?? 2
 T
-2
 A
-1
]
 
SI UNIT of magnetic flux: 
? [ML
2
 T
-2
] Corresponds to energy 
 joule 
 ampere 
=
 joule × second 
 coulomb 
= weber (Wb) 
Or  T-m
2
( as tesla =Wb/m
2
) [ ampere =
 coulomb 
 sec ond 
] 
? For a given area flux will be maximum: 
 
When magnetic field ??? 
 is normal to the area 
?? =0
°
 ?cos ?? =maximum=1 ?? max
=BA 
For a given area flux will be minimum: 
When magnetic field ??? 
 is parallel to the area 
 
?? =90
°
?cos ?? = minimum =0 ?? min
=0 
Example: Find flux associated with given areas. 
1. 
 
3. 
 
4. 
 
5. 
 
??ˆ
? 
?? =?? ?? ?? 2
?? =0
 
??ˆ
=? 
?? =-B
(Lx)
2
 
?? =BAcos 60
°
=
BA
2
 
?? =BAcos 120
°
=-
BA
2
 
6. 
 
B
0
?? i
=-BA
?? i
=BA
??? =2BA
A
ˆ
=?
|?? b
|=|?? f
-?? i
 
Example. A loop of area 0.06 m
2
 is placed in a magnetic field 1.2 T with its plane 
inclined 30
°
 to the field direction. Find the flux linked with plane of loop. 
Solution: Area of loop is 0.06 m
2
, B=1.2 T and ?? =90
°
-30
°
=60
°
 
So, the flux linked with the loop is 
?? =BAcos ?? =1.2×0.06×cos ?? =1.2×0.06×
1
2
??? =0.036 Wb 
Example. At a given plane, horizontal and vertical components of earth's magnetic field 
B
H
 and B
V
 are along x and y axes respectively as shown in figure. What is the total flux of 
earth's magnetic field associated with an area ?? , if the area ?? is in (a) ?? -?? plane (b) ?? -
?? plane and (c) ?? -?? plane? 
 
 
Solution: : ??? 
=??ˆ?? ?? -??ˆ?? ?? = constant, so ?? =??? 
·?? 
[??? 
= constant ] 
(a) For area in ?? -?? plane ?? 
=?? ??ˆ
,?? ????
=(??ˆ?? ?? -??ˆ?? ?? )·(??ˆ
?? )=0 
(b) For area ?? in ?? -?? plane ?? 
=?? ??ˆ,?? ????
=(??ˆ?? ?? -??ˆ?? ?? )·(??ˆ?? )=?? ?? ?? 
(c) For area ?? in ?? -?? plane ?? 
=?? ??ˆ,?? ????
=(??ˆ?? ?? -??ˆ?? ?? )·(??ˆ?? )=-?? ?? ?? 
A negative sign implies that flux is directed vertically downwards. 
Example 
 
Solution: ?? =
?? 0
i
2?? x
 
 ? d?? =? ?
1+a
l
?
?? 0
i
2?? x
bdx
?? i
=
?? i
2?? bln(
l+a
l
)
 ? d?? =? ?
l+a
?
?? 0
i
2?? x
bdxcos 180
°
?? f
=
-?? 0
ib
2?? ln(
l+2a
l+a
)
??? =?? f
-?? i
=
?? 0
i
2?? b[-ln(
l+2a
l+a
)-ln(
l+a
l
)]
 =-
?? 0
 b
2?? ln (
l+2a
l
)=-
?? 0
ib
2?? ln (
l+2a
l
)
 
(b) If shifted without rotating to same final position ??? = ? 
Page 5


ELECTROMAGNETIC INDUCTION (EMI) 
INTRODUCTION 
 
In 1820, Oersted found out that electric current makes magnets move. Later, Faraday 
had an idea: what if it works the other way too? What if magnets can make electric 
current? Faraday tested this by moving a magnet near a coil of wire connected to a 
sensitive device called a galvanometer. When the magnet moved through the coil, the 
galvanometer immediately showed a change.  
Magnetic Flux (?? ) : 
Magnetic flux through an area ?? is given by ?? 
?? =???? 
·?? ?? 
 ??? 
= magnetic field intensity 
 
SI unit of ?? is Weber (Wb) or Tesla m
2
. 
If B
?? 
 is constant 
?? =B
?? 
·A
?? 
 A
?? 
= Area vector whose direction is normal to the plane of area. 
?? =|B
?? 
|·|A
?? 
|cos ?? 
Note: Any one the two normal can be taken as direction of area vector but throughout a 
single calculation conversation taken should not be charged. 
FLUX LINKAGE 
If a coil has more than one turn, then the flux through the whole coil is the sum of the 
fluxes through the individual turns. If the magnetic field is uniform, the flux through one 
turn is ?? =BAcos ?? If the coil has N turns, the total flux linkage ?? = NBA cos ?? 
? Magnetic lines of force are imaginary, magnetic flux is a real scalar physical 
quantity with dimensions [?? ]=B× area =[
F
IL
][L
2
] 
? B=
F
ILsin ?? ? [?? ]=[
???? ?? -2
AL
][L
2
]=[?? ?? 2
 T
-2
 A
-1
]
 
SI UNIT of magnetic flux: 
? [ML
2
 T
-2
] Corresponds to energy 
 joule 
 ampere 
=
 joule × second 
 coulomb 
= weber (Wb) 
Or  T-m
2
( as tesla =Wb/m
2
) [ ampere =
 coulomb 
 sec ond 
] 
? For a given area flux will be maximum: 
 
When magnetic field ??? 
 is normal to the area 
?? =0
°
 ?cos ?? =maximum=1 ?? max
=BA 
For a given area flux will be minimum: 
When magnetic field ??? 
 is parallel to the area 
 
?? =90
°
?cos ?? = minimum =0 ?? min
=0 
Example: Find flux associated with given areas. 
1. 
 
3. 
 
4. 
 
5. 
 
??ˆ
? 
?? =?? ?? ?? 2
?? =0
 
??ˆ
=? 
?? =-B
(Lx)
2
 
?? =BAcos 60
°
=
BA
2
 
?? =BAcos 120
°
=-
BA
2
 
6. 
 
B
0
?? i
=-BA
?? i
=BA
??? =2BA
A
ˆ
=?
|?? b
|=|?? f
-?? i
 
Example. A loop of area 0.06 m
2
 is placed in a magnetic field 1.2 T with its plane 
inclined 30
°
 to the field direction. Find the flux linked with plane of loop. 
Solution: Area of loop is 0.06 m
2
, B=1.2 T and ?? =90
°
-30
°
=60
°
 
So, the flux linked with the loop is 
?? =BAcos ?? =1.2×0.06×cos ?? =1.2×0.06×
1
2
??? =0.036 Wb 
Example. At a given plane, horizontal and vertical components of earth's magnetic field 
B
H
 and B
V
 are along x and y axes respectively as shown in figure. What is the total flux of 
earth's magnetic field associated with an area ?? , if the area ?? is in (a) ?? -?? plane (b) ?? -
?? plane and (c) ?? -?? plane? 
 
 
Solution: : ??? 
=??ˆ?? ?? -??ˆ?? ?? = constant, so ?? =??? 
·?? 
[??? 
= constant ] 
(a) For area in ?? -?? plane ?? 
=?? ??ˆ
,?? ????
=(??ˆ?? ?? -??ˆ?? ?? )·(??ˆ
?? )=0 
(b) For area ?? in ?? -?? plane ?? 
=?? ??ˆ,?? ????
=(??ˆ?? ?? -??ˆ?? ?? )·(??ˆ?? )=?? ?? ?? 
(c) For area ?? in ?? -?? plane ?? 
=?? ??ˆ,?? ????
=(??ˆ?? ?? -??ˆ?? ?? )·(??ˆ?? )=-?? ?? ?? 
A negative sign implies that flux is directed vertically downwards. 
Example 
 
Solution: ?? =
?? 0
i
2?? x
 
 ? d?? =? ?
1+a
l
?
?? 0
i
2?? x
bdx
?? i
=
?? i
2?? bln(
l+a
l
)
 ? d?? =? ?
l+a
?
?? 0
i
2?? x
bdxcos 180
°
?? f
=
-?? 0
ib
2?? ln(
l+2a
l+a
)
??? =?? f
-?? i
=
?? 0
i
2?? b[-ln(
l+2a
l+a
)-ln(
l+a
l
)]
 =-
?? 0
 b
2?? ln (
l+2a
l
)=-
?? 0
ib
2?? ln (
l+2a
l
)
 
(b) If shifted without rotating to same final position ??? = ? 
?? f
=
?? 0
ib
2?? ln(
l+2a
l+a
)cos 0
°
 (?? will remain same )
??? =
?? 0
ib
2?? [ln (
l+2a
l+a
)-ln (
l+a
l
)]=
?? 0
ib
2?? ln (
l(l+2a)
(l+a)
2
)
 
Example. 
 
A magnetic field perpendicular to ?? -?? plane is given as ?? =?? 0
(
?? ?? )(-??ˆ
) [independent 
of l as ?? has linear variation of distance] 
1. Find flux associated with square of side a as show in the figure 
2. Find 
d?? dt
 if loop is moving with velocity v in positive ?? -direction. 
Solution: ???? =?
?? 0
?? ?? ×(adx) 
?? =B
0
xdx=
B
0
x
2
2
?? =
B
0
x
2
2
|
l
?? +a
=B
0
(
(l+a)
2
2
-l
2
)
d?? dt
=
B
0
2
[2(l+a)
dl
dt
-2l
dl
dt
]=
B
0
2
[2(l+a)v-2lv]=B
0
[av]=avB
0
=B
0
va
 OR 
 
 
???? =
?? 0
?? [(?? +?? )-???? ]???????? 
Example. A loop of area a is rotating about its side as shown in the figure in a magnetic 
field perpendicular to plane, find 
d?? dt
 
 
Solution: t=0; ?? =0