Magnetic Effects of Current | Physics Optional Notes for UPSC PDF Download

 Page 1


MAGNETIC EFFECT OF CURRENT 
LECTURES NOTES 
Current Element : 
 
A current element is a tiny part (like a small segment) of a thin conductor that carries an 
electric a current I. It's considered a vector quantity because it has both magnitude and 
direction. The magnitude of the current element is found by multiplying the current I by 
the length dl of the small segment, and its direction aligns with the flow of current through 
the conductor. 
Biot- Savart's Law: 
Biot and Savart conducted experiments to understand how the magnetic field is produced 
by a current-carrying wire. They came up with a mathematical formula that relates the 
magnetic field at a point in space to the current in the wire. This formula is based on their 
experimental observations regarding the magnetic field dB at a point P caused by a tiny 
length element dl of a wire carrying a constant current I. 
dB? I,dB? dl,dB? sin ??? and dB?
1
r
2
? ?dB?
Idlsin ??? r
2
? dB=
?? 0
4?? Idlsin ??? r
2
 
Where ?? 0
 a constant is called the permeability of free space: 
?? 0
= 4?? × 10
-7
 T.m/A here Tesla (T) is SI unit of B
? ? 
 
Vector form of Biot-Savart's law: 
dB
? ? 
=
?? 0
4?? Id?lsin??? r
2
n ˆ?n ˆ = Unit vector perpendicular to the plane of (Id?l
? 
) and (r ) 
dB
? ? 
=
?? 0
4?? ddl
? 
× r 
r
3
?[? Id?l
? 
× r = (Id?l)(r)sin ??? n ˆ] 
KEY POINTS 
? According to?dB
? ? 
=
?? 0
4?? Idl
? 
×r ? 
r
3
, direction of magnetic field vector (dB
? ? 
 ) is always 
perpendicular to the plane of vectors (Id?l
? 
) and (r ) , where plane of (Id?l
? 
) and (r ) 
is the plane of wire. 
Page 2


MAGNETIC EFFECT OF CURRENT 
LECTURES NOTES 
Current Element : 
 
A current element is a tiny part (like a small segment) of a thin conductor that carries an 
electric a current I. It's considered a vector quantity because it has both magnitude and 
direction. The magnitude of the current element is found by multiplying the current I by 
the length dl of the small segment, and its direction aligns with the flow of current through 
the conductor. 
Biot- Savart's Law: 
Biot and Savart conducted experiments to understand how the magnetic field is produced 
by a current-carrying wire. They came up with a mathematical formula that relates the 
magnetic field at a point in space to the current in the wire. This formula is based on their 
experimental observations regarding the magnetic field dB at a point P caused by a tiny 
length element dl of a wire carrying a constant current I. 
dB? I,dB? dl,dB? sin ??? and dB?
1
r
2
? ?dB?
Idlsin ??? r
2
? dB=
?? 0
4?? Idlsin ??? r
2
 
Where ?? 0
 a constant is called the permeability of free space: 
?? 0
= 4?? × 10
-7
 T.m/A here Tesla (T) is SI unit of B
? ? 
 
Vector form of Biot-Savart's law: 
dB
? ? 
=
?? 0
4?? Id?lsin??? r
2
n ˆ?n ˆ = Unit vector perpendicular to the plane of (Id?l
? 
) and (r ) 
dB
? ? 
=
?? 0
4?? ddl
? 
× r 
r
3
?[? Id?l
? 
× r = (Id?l)(r)sin ??? n ˆ] 
KEY POINTS 
? According to?dB
? ? 
=
?? 0
4?? Idl
? 
×r ? 
r
3
, direction of magnetic field vector (dB
? ? 
 ) is always 
perpendicular to the plane of vectors (Id?l
? 
) and (r ) , where plane of (Id?l
? 
) and (r ) 
is the plane of wire. 
? Magnetic field on the axis of current carrying conductor is always zero (?? = 0
°
 or 
?? = 180
°
) 
? Magnetic field on the perimeter of circular loop or coil is always minimum. 
RIGHT HAND THUMB RULE 
This rule gives the pattern of magnetic field lines due to current carrying wire. 
(i) Straight current 
Thumb ? In the direction of current Curling fingers ? Gives field line pattern  
Case I : wire in the plane of the paper (ii) Circular current Curling fingers ? In the 
direction of current, Thumb ? Gives field line pattern Case 1: wire in the plane of the 
paper 
 
Case II : Wire is ? to the plane of the paper. 
 
 
Case II : Wire is ? to the plane of the paper 
 
KEY POINTS 
? When current is straight, field is circular 
? When current is circular, field is straight (along axis) 
Page 3


MAGNETIC EFFECT OF CURRENT 
LECTURES NOTES 
Current Element : 
 
A current element is a tiny part (like a small segment) of a thin conductor that carries an 
electric a current I. It's considered a vector quantity because it has both magnitude and 
direction. The magnitude of the current element is found by multiplying the current I by 
the length dl of the small segment, and its direction aligns with the flow of current through 
the conductor. 
Biot- Savart's Law: 
Biot and Savart conducted experiments to understand how the magnetic field is produced 
by a current-carrying wire. They came up with a mathematical formula that relates the 
magnetic field at a point in space to the current in the wire. This formula is based on their 
experimental observations regarding the magnetic field dB at a point P caused by a tiny 
length element dl of a wire carrying a constant current I. 
dB? I,dB? dl,dB? sin ??? and dB?
1
r
2
? ?dB?
Idlsin ??? r
2
? dB=
?? 0
4?? Idlsin ??? r
2
 
Where ?? 0
 a constant is called the permeability of free space: 
?? 0
= 4?? × 10
-7
 T.m/A here Tesla (T) is SI unit of B
? ? 
 
Vector form of Biot-Savart's law: 
dB
? ? 
=
?? 0
4?? Id?lsin??? r
2
n ˆ?n ˆ = Unit vector perpendicular to the plane of (Id?l
? 
) and (r ) 
dB
? ? 
=
?? 0
4?? ddl
? 
× r 
r
3
?[? Id?l
? 
× r = (Id?l)(r)sin ??? n ˆ] 
KEY POINTS 
? According to?dB
? ? 
=
?? 0
4?? Idl
? 
×r ? 
r
3
, direction of magnetic field vector (dB
? ? 
 ) is always 
perpendicular to the plane of vectors (Id?l
? 
) and (r ) , where plane of (Id?l
? 
) and (r ) 
is the plane of wire. 
? Magnetic field on the axis of current carrying conductor is always zero (?? = 0
°
 or 
?? = 180
°
) 
? Magnetic field on the perimeter of circular loop or coil is always minimum. 
RIGHT HAND THUMB RULE 
This rule gives the pattern of magnetic field lines due to current carrying wire. 
(i) Straight current 
Thumb ? In the direction of current Curling fingers ? Gives field line pattern  
Case I : wire in the plane of the paper (ii) Circular current Curling fingers ? In the 
direction of current, Thumb ? Gives field line pattern Case 1: wire in the plane of the 
paper 
 
Case II : Wire is ? to the plane of the paper. 
 
 
Case II : Wire is ? to the plane of the paper 
 
KEY POINTS 
? When current is straight, field is circular 
? When current is circular, field is straight (along axis) 
? When wire is in the plane of paper, the field is perpendicular to the plane of the 
paper. 
? When wire is perpendicular to the plane of paper, the field is in the plane of the 
paper. 
APPLICATION OF BIOT-SAVART LAW: 
? Magnetic field surrounding a thin straight current carrying conductor 
 
 
???? is a straight conductor carrying current i from ?? to ?? . 
At a point P, whose perpendicular distance from AB is OP = a, the direction of field is 
perpendicular to the plane of paper, inwards (represented by a cross) 
l = atan ??? ? d?? = asec
2
?? d?? … (i) 
?? = 90
°
- ?? &?? = asec ??? 
- By Biot-Savart's law: 
dB
????? 
=
?? 0
4?? idlsin??? r
2
? ( due to a current element idl at point P) 
? ?? = ????? = ?
?? 0
4?? idlsin??? ?? 2
 (due to wire ???? ) ? ?? =
?? 0
i
4?? ?cos ??????? 
Taking limits of integration as -?? 2
 to ?? 1
 
?? =
?? 0
i
4?? a
?
-?? 2
?? 1
?cos ??? d?? =
?? 0
i
4?? a
[sin ??? ]
-?? 2
?? 1
=
?? 0
i
4?? a
[sin ??? 1
+ sin ??? 2
] (inwards) 
HOW TO TAKE LIMIT IN ?? =
?? 0
?? 4????
?cos ??????? ??? is perpendicular distance of ?? from wire. 
(a) 
 
?? =
?? 0
i
4?? R
? ?
?? 1
-?? 2
cos ??? d?? 
Page 4


MAGNETIC EFFECT OF CURRENT 
LECTURES NOTES 
Current Element : 
 
A current element is a tiny part (like a small segment) of a thin conductor that carries an 
electric a current I. It's considered a vector quantity because it has both magnitude and 
direction. The magnitude of the current element is found by multiplying the current I by 
the length dl of the small segment, and its direction aligns with the flow of current through 
the conductor. 
Biot- Savart's Law: 
Biot and Savart conducted experiments to understand how the magnetic field is produced 
by a current-carrying wire. They came up with a mathematical formula that relates the 
magnetic field at a point in space to the current in the wire. This formula is based on their 
experimental observations regarding the magnetic field dB at a point P caused by a tiny 
length element dl of a wire carrying a constant current I. 
dB? I,dB? dl,dB? sin ??? and dB?
1
r
2
? ?dB?
Idlsin ??? r
2
? dB=
?? 0
4?? Idlsin ??? r
2
 
Where ?? 0
 a constant is called the permeability of free space: 
?? 0
= 4?? × 10
-7
 T.m/A here Tesla (T) is SI unit of B
? ? 
 
Vector form of Biot-Savart's law: 
dB
? ? 
=
?? 0
4?? Id?lsin??? r
2
n ˆ?n ˆ = Unit vector perpendicular to the plane of (Id?l
? 
) and (r ) 
dB
? ? 
=
?? 0
4?? ddl
? 
× r 
r
3
?[? Id?l
? 
× r = (Id?l)(r)sin ??? n ˆ] 
KEY POINTS 
? According to?dB
? ? 
=
?? 0
4?? Idl
? 
×r ? 
r
3
, direction of magnetic field vector (dB
? ? 
 ) is always 
perpendicular to the plane of vectors (Id?l
? 
) and (r ) , where plane of (Id?l
? 
) and (r ) 
is the plane of wire. 
? Magnetic field on the axis of current carrying conductor is always zero (?? = 0
°
 or 
?? = 180
°
) 
? Magnetic field on the perimeter of circular loop or coil is always minimum. 
RIGHT HAND THUMB RULE 
This rule gives the pattern of magnetic field lines due to current carrying wire. 
(i) Straight current 
Thumb ? In the direction of current Curling fingers ? Gives field line pattern  
Case I : wire in the plane of the paper (ii) Circular current Curling fingers ? In the 
direction of current, Thumb ? Gives field line pattern Case 1: wire in the plane of the 
paper 
 
Case II : Wire is ? to the plane of the paper. 
 
 
Case II : Wire is ? to the plane of the paper 
 
KEY POINTS 
? When current is straight, field is circular 
? When current is circular, field is straight (along axis) 
? When wire is in the plane of paper, the field is perpendicular to the plane of the 
paper. 
? When wire is perpendicular to the plane of paper, the field is in the plane of the 
paper. 
APPLICATION OF BIOT-SAVART LAW: 
? Magnetic field surrounding a thin straight current carrying conductor 
 
 
???? is a straight conductor carrying current i from ?? to ?? . 
At a point P, whose perpendicular distance from AB is OP = a, the direction of field is 
perpendicular to the plane of paper, inwards (represented by a cross) 
l = atan ??? ? d?? = asec
2
?? d?? … (i) 
?? = 90
°
- ?? &?? = asec ??? 
- By Biot-Savart's law: 
dB
????? 
=
?? 0
4?? idlsin??? r
2
? ( due to a current element idl at point P) 
? ?? = ????? = ?
?? 0
4?? idlsin??? ?? 2
 (due to wire ???? ) ? ?? =
?? 0
i
4?? ?cos ??????? 
Taking limits of integration as -?? 2
 to ?? 1
 
?? =
?? 0
i
4?? a
?
-?? 2
?? 1
?cos ??? d?? =
?? 0
i
4?? a
[sin ??? ]
-?? 2
?? 1
=
?? 0
i
4?? a
[sin ??? 1
+ sin ??? 2
] (inwards) 
HOW TO TAKE LIMIT IN ?? =
?? 0
?? 4????
?cos ??????? ??? is perpendicular distance of ?? from wire. 
(a) 
 
?? =
?? 0
i
4?? R
? ?
?? 1
-?? 2
cos ??? d?? 
(b) 
 
 
Perpendicular bisector: 
 
?? =
?? 0
?? 4????
? ?
?? -?? cos ??????? ? ? ??? =
?? 0
?? sin ??? 2????
 
Discuss the following cases: 
(i) In the symmetric case where??? 2
= -?? 1
, the field point ?? is located along the 
perpendicular bisector. If the length of the rod is 2?? , then sin ??? 1
= L/vL
2
+ a
2
 and the 
magnetic field is 
?? =
?? 0
?? 2?? a
sin ??? 1
=
?? 0
?? 2????
?? v?? 2
+ ?? 2
 
MFI due to 8 wire : 
 
?? =
?? 0
i
4?? R
? ?
?? /2
-?? /2
cos ??? d?? =
?? 0
i(2)
4?? R
=
?? 0
i
2?? R
? B =
?? 0
i
2?? R
 
 
In fact, the direction of the magnetic field due to a long straight wire can be determined 
by the right-hand rule (Figure). 
Page 5


MAGNETIC EFFECT OF CURRENT 
LECTURES NOTES 
Current Element : 
 
A current element is a tiny part (like a small segment) of a thin conductor that carries an 
electric a current I. It's considered a vector quantity because it has both magnitude and 
direction. The magnitude of the current element is found by multiplying the current I by 
the length dl of the small segment, and its direction aligns with the flow of current through 
the conductor. 
Biot- Savart's Law: 
Biot and Savart conducted experiments to understand how the magnetic field is produced 
by a current-carrying wire. They came up with a mathematical formula that relates the 
magnetic field at a point in space to the current in the wire. This formula is based on their 
experimental observations regarding the magnetic field dB at a point P caused by a tiny 
length element dl of a wire carrying a constant current I. 
dB? I,dB? dl,dB? sin ??? and dB?
1
r
2
? ?dB?
Idlsin ??? r
2
? dB=
?? 0
4?? Idlsin ??? r
2
 
Where ?? 0
 a constant is called the permeability of free space: 
?? 0
= 4?? × 10
-7
 T.m/A here Tesla (T) is SI unit of B
? ? 
 
Vector form of Biot-Savart's law: 
dB
? ? 
=
?? 0
4?? Id?lsin??? r
2
n ˆ?n ˆ = Unit vector perpendicular to the plane of (Id?l
? 
) and (r ) 
dB
? ? 
=
?? 0
4?? ddl
? 
× r 
r
3
?[? Id?l
? 
× r = (Id?l)(r)sin ??? n ˆ] 
KEY POINTS 
? According to?dB
? ? 
=
?? 0
4?? Idl
? 
×r ? 
r
3
, direction of magnetic field vector (dB
? ? 
 ) is always 
perpendicular to the plane of vectors (Id?l
? 
) and (r ) , where plane of (Id?l
? 
) and (r ) 
is the plane of wire. 
? Magnetic field on the axis of current carrying conductor is always zero (?? = 0
°
 or 
?? = 180
°
) 
? Magnetic field on the perimeter of circular loop or coil is always minimum. 
RIGHT HAND THUMB RULE 
This rule gives the pattern of magnetic field lines due to current carrying wire. 
(i) Straight current 
Thumb ? In the direction of current Curling fingers ? Gives field line pattern  
Case I : wire in the plane of the paper (ii) Circular current Curling fingers ? In the 
direction of current, Thumb ? Gives field line pattern Case 1: wire in the plane of the 
paper 
 
Case II : Wire is ? to the plane of the paper. 
 
 
Case II : Wire is ? to the plane of the paper 
 
KEY POINTS 
? When current is straight, field is circular 
? When current is circular, field is straight (along axis) 
? When wire is in the plane of paper, the field is perpendicular to the plane of the 
paper. 
? When wire is perpendicular to the plane of paper, the field is in the plane of the 
paper. 
APPLICATION OF BIOT-SAVART LAW: 
? Magnetic field surrounding a thin straight current carrying conductor 
 
 
???? is a straight conductor carrying current i from ?? to ?? . 
At a point P, whose perpendicular distance from AB is OP = a, the direction of field is 
perpendicular to the plane of paper, inwards (represented by a cross) 
l = atan ??? ? d?? = asec
2
?? d?? … (i) 
?? = 90
°
- ?? &?? = asec ??? 
- By Biot-Savart's law: 
dB
????? 
=
?? 0
4?? idlsin??? r
2
? ( due to a current element idl at point P) 
? ?? = ????? = ?
?? 0
4?? idlsin??? ?? 2
 (due to wire ???? ) ? ?? =
?? 0
i
4?? ?cos ??????? 
Taking limits of integration as -?? 2
 to ?? 1
 
?? =
?? 0
i
4?? a
?
-?? 2
?? 1
?cos ??? d?? =
?? 0
i
4?? a
[sin ??? ]
-?? 2
?? 1
=
?? 0
i
4?? a
[sin ??? 1
+ sin ??? 2
] (inwards) 
HOW TO TAKE LIMIT IN ?? =
?? 0
?? 4????
?cos ??????? ??? is perpendicular distance of ?? from wire. 
(a) 
 
?? =
?? 0
i
4?? R
? ?
?? 1
-?? 2
cos ??? d?? 
(b) 
 
 
Perpendicular bisector: 
 
?? =
?? 0
?? 4????
? ?
?? -?? cos ??????? ? ? ??? =
?? 0
?? sin ??? 2????
 
Discuss the following cases: 
(i) In the symmetric case where??? 2
= -?? 1
, the field point ?? is located along the 
perpendicular bisector. If the length of the rod is 2?? , then sin ??? 1
= L/vL
2
+ a
2
 and the 
magnetic field is 
?? =
?? 0
?? 2?? a
sin ??? 1
=
?? 0
?? 2????
?? v?? 2
+ ?? 2
 
MFI due to 8 wire : 
 
?? =
?? 0
i
4?? R
? ?
?? /2
-?? /2
cos ??? d?? =
?? 0
i(2)
4?? R
=
?? 0
i
2?? R
? B =
?? 0
i
2?? R
 
 
In fact, the direction of the magnetic field due to a long straight wire can be determined 
by the right-hand rule (Figure). 
 
Figure: Direction of the magnetic field due to an infinite straight wire 
 
{Note: that in this limit, the system possesses cylindrical symmetry, and the magnetic 
field lines are circular.} 
Direction of MFI due to straight wire: 
 
 
? 
 
? In this case magnetic field lines are circular. 
? Hold the wire with right hand with thumb pointing along current then direction of 
curling of finger will give direction of MFI . 
Example. Magnetic field due to semi-infinite length wire at point ' P ' 
Solution: ?? ?? =
?? 0
?? 4????
?
-?? 90
°
?cos ??????? 
B
P
=
?? 0
I
4?? d
[sin ??? + sin ?90
°
]
B
P
=
?? 0
I
4?? d
[sin ??? + 1]