Magnetic effect of current for IIT JEE Class 12 Notes | EduRev

Class 12 : Magnetic effect of current for IIT JEE Class 12 Notes | EduRev

 Page 1


 
 
Magnetic Effect of Current 1 
1
genius   PHYSICS 
 
 
 
 
 
 
Oersted found that a magnetic field is established around a current 
carrying conductor. 
Magnetic field exists as long as there is current in the wire. 
The direction of magnetic field was found to be changed when direction of 
current was reversed. 
Note : ? A moving charge produces magnetic as well as electric field, 
unlike a stationary charge which only produces electric field.  
 Biot Savart's Law. 
Biot-Savart’s law is used to determine the magnetic field at any point due to a current carrying 
conductors. 
This law is although for infinitesimally small conductors yet it can be used for long conductors. In order to 
understand the Biot-Savart’s law, we need to understand the term current-element. 
 
Current element  
It is the product of current and length of infinitesimal segment of current carrying wire. 
The current element is taken as a vector quantity. Its direction is same as the direction of current. 
Current element AB = dl i 
 
In the figure shown below, there is a segment of current carrying wire and P is a point where magnetic 
field is to be calculated. l d i is a current element and r is the distance of the point ‘P’ with respect to the current 
element l d i . According to Biot-Savart Law, magnetic field at point ‘P’ due to the current element l d i is given 
by the expression, 
2
r
dlsin ? i
k dB ? also 
? ?
? ?
2
0
sin
.
4 r
dl i
dB B
?
?
?
 
In C.G.S. : k = 1 ? 
2
sin
r
idl
dB
?
? Gauss  
In S.I. : 
?
?
4
0
? k ? 
2
0
sin
4 r
idl
dB
?
?
?
? ? Tesla  
where 
0
? = Absolute permeability of air or vacuum 
metre Amp
Wb
?
? ?
?7
10 4 ? . It's other units are 
metre
Henry
  
or 
2
Amp
N
 or 
Ampere
metre Tesla ?
 
 
 
dl 
i 
P 
r 
A 
B 
dl 
i 
Magnetic lines 
of forces 
i 
Page 2


 
 
Magnetic Effect of Current 1 
1
genius   PHYSICS 
 
 
 
 
 
 
Oersted found that a magnetic field is established around a current 
carrying conductor. 
Magnetic field exists as long as there is current in the wire. 
The direction of magnetic field was found to be changed when direction of 
current was reversed. 
Note : ? A moving charge produces magnetic as well as electric field, 
unlike a stationary charge which only produces electric field.  
 Biot Savart's Law. 
Biot-Savart’s law is used to determine the magnetic field at any point due to a current carrying 
conductors. 
This law is although for infinitesimally small conductors yet it can be used for long conductors. In order to 
understand the Biot-Savart’s law, we need to understand the term current-element. 
 
Current element  
It is the product of current and length of infinitesimal segment of current carrying wire. 
The current element is taken as a vector quantity. Its direction is same as the direction of current. 
Current element AB = dl i 
 
In the figure shown below, there is a segment of current carrying wire and P is a point where magnetic 
field is to be calculated. l d i is a current element and r is the distance of the point ‘P’ with respect to the current 
element l d i . According to Biot-Savart Law, magnetic field at point ‘P’ due to the current element l d i is given 
by the expression, 
2
r
dlsin ? i
k dB ? also 
? ?
? ?
2
0
sin
.
4 r
dl i
dB B
?
?
?
 
In C.G.S. : k = 1 ? 
2
sin
r
idl
dB
?
? Gauss  
In S.I. : 
?
?
4
0
? k ? 
2
0
sin
4 r
idl
dB
?
?
?
? ? Tesla  
where 
0
? = Absolute permeability of air or vacuum 
metre Amp
Wb
?
? ?
?7
10 4 ? . It's other units are 
metre
Henry
  
or 
2
Amp
N
 or 
Ampere
metre Tesla ?
 
 
 
dl 
i 
P 
r 
A 
B 
dl 
i 
Magnetic lines 
of forces 
i 
 
 
2 Magnetic Effect of Current 
genius   PHYSICS 
 
(1) Different forms of Biot-Savarts law  
Vector form Biot-Savarts law in terms of 
current density 
Biot-savarts law in terms of 
charge and it's velocity 
Vectorially, 
3
0
2
0
) (
4
)
ˆ
(
4 r
r l d i
r
r l d i
B d
?
? ?
?
? ?
? ?
?
?
?
?
?
 ? 
Direction of B d
?
 is perpendicular to 
both l d
?
 and r
ˆ . This is given by 
right hand screw rule. 
 
In terms of current density 
dV
r
r J
B d
3
0
4
?
?
?
?
?
?
?
 
where 
dV
idl
Adl
idl
A
i
j ? ? ? = current 
density at any point of the element, 
dV = volume of element 
In terms of charge and it’s velocity, 
3
0
) (
4
r
r v
q B d
? ?
?
?
?
?
?
 
? v q
dt
l d
q l d
dt
q
l id
?
?
? ?
? ? ? 
 
(2) Similarities and differences between Biot-Savart law and Coulomb’s Law 
(i) The current element produces a magnetic field, whereas a point charge produces an electric field. 
(ii) The magnitude of magnetic field varies as the inverse square of the distance from the current element, 
as does the electric field due to a point charge. 
  
2
0
ˆ
4 r
r l id
B d
?
?
?
?
 Biot-Savart Law r
r
q q
F ˆ
4
1
2
2 1
0
??
? Coulomb’s Law 
(iii) The electric field created by a point charge is radial, but the magnetic field created by a current 
element is perpendicular to both the length element l d and the unit vector r ˆ . 
 
 
 
 
 Direction of Magnetic Field. 
The direction of magnetic field is determined with the help of the following simple laws : 
(1) Maxwell’s cork screw rule 
According to this rule, if we imagine a right handed screw placed along the current 
carrying linear conductor, be rotated such that the screw moves in the direction of flow of 
current, then the direction of rotation of the thumb gives the direction of magnetic lines of 
force.  
 
(2) Right hand thumb rule 
According to this rule if a current carrying conductor is held in the right hand such 
that the thumb of the hand represents the direction of current flow, then the direction of 
folding fingers will represent the direction of magnetic lines of force. 
 
(3) Right hand thumb rule of circular currents 
According to this rule if the direction of current in circular 
B 
i 
B 
dl 
i 
r
ˆ 
B 
r ˆ
 +q 
E 
Page 3


 
 
Magnetic Effect of Current 1 
1
genius   PHYSICS 
 
 
 
 
 
 
Oersted found that a magnetic field is established around a current 
carrying conductor. 
Magnetic field exists as long as there is current in the wire. 
The direction of magnetic field was found to be changed when direction of 
current was reversed. 
Note : ? A moving charge produces magnetic as well as electric field, 
unlike a stationary charge which only produces electric field.  
 Biot Savart's Law. 
Biot-Savart’s law is used to determine the magnetic field at any point due to a current carrying 
conductors. 
This law is although for infinitesimally small conductors yet it can be used for long conductors. In order to 
understand the Biot-Savart’s law, we need to understand the term current-element. 
 
Current element  
It is the product of current and length of infinitesimal segment of current carrying wire. 
The current element is taken as a vector quantity. Its direction is same as the direction of current. 
Current element AB = dl i 
 
In the figure shown below, there is a segment of current carrying wire and P is a point where magnetic 
field is to be calculated. l d i is a current element and r is the distance of the point ‘P’ with respect to the current 
element l d i . According to Biot-Savart Law, magnetic field at point ‘P’ due to the current element l d i is given 
by the expression, 
2
r
dlsin ? i
k dB ? also 
? ?
? ?
2
0
sin
.
4 r
dl i
dB B
?
?
?
 
In C.G.S. : k = 1 ? 
2
sin
r
idl
dB
?
? Gauss  
In S.I. : 
?
?
4
0
? k ? 
2
0
sin
4 r
idl
dB
?
?
?
? ? Tesla  
where 
0
? = Absolute permeability of air or vacuum 
metre Amp
Wb
?
? ?
?7
10 4 ? . It's other units are 
metre
Henry
  
or 
2
Amp
N
 or 
Ampere
metre Tesla ?
 
 
 
dl 
i 
P 
r 
A 
B 
dl 
i 
Magnetic lines 
of forces 
i 
 
 
2 Magnetic Effect of Current 
genius   PHYSICS 
 
(1) Different forms of Biot-Savarts law  
Vector form Biot-Savarts law in terms of 
current density 
Biot-savarts law in terms of 
charge and it's velocity 
Vectorially, 
3
0
2
0
) (
4
)
ˆ
(
4 r
r l d i
r
r l d i
B d
?
? ?
?
? ?
? ?
?
?
?
?
?
 ? 
Direction of B d
?
 is perpendicular to 
both l d
?
 and r
ˆ . This is given by 
right hand screw rule. 
 
In terms of current density 
dV
r
r J
B d
3
0
4
?
?
?
?
?
?
?
 
where 
dV
idl
Adl
idl
A
i
j ? ? ? = current 
density at any point of the element, 
dV = volume of element 
In terms of charge and it’s velocity, 
3
0
) (
4
r
r v
q B d
? ?
?
?
?
?
?
 
? v q
dt
l d
q l d
dt
q
l id
?
?
? ?
? ? ? 
 
(2) Similarities and differences between Biot-Savart law and Coulomb’s Law 
(i) The current element produces a magnetic field, whereas a point charge produces an electric field. 
(ii) The magnitude of magnetic field varies as the inverse square of the distance from the current element, 
as does the electric field due to a point charge. 
  
2
0
ˆ
4 r
r l id
B d
?
?
?
?
 Biot-Savart Law r
r
q q
F ˆ
4
1
2
2 1
0
??
? Coulomb’s Law 
(iii) The electric field created by a point charge is radial, but the magnetic field created by a current 
element is perpendicular to both the length element l d and the unit vector r ˆ . 
 
 
 
 
 Direction of Magnetic Field. 
The direction of magnetic field is determined with the help of the following simple laws : 
(1) Maxwell’s cork screw rule 
According to this rule, if we imagine a right handed screw placed along the current 
carrying linear conductor, be rotated such that the screw moves in the direction of flow of 
current, then the direction of rotation of the thumb gives the direction of magnetic lines of 
force.  
 
(2) Right hand thumb rule 
According to this rule if a current carrying conductor is held in the right hand such 
that the thumb of the hand represents the direction of current flow, then the direction of 
folding fingers will represent the direction of magnetic lines of force. 
 
(3) Right hand thumb rule of circular currents 
According to this rule if the direction of current in circular 
B 
i 
B 
dl 
i 
r
ˆ 
B 
r ˆ
 +q 
E 
 
 
Magnetic Effect of Current 3 
3
genius   PHYSICS 
conducting coil is in the direction of folding fingers of right hand, then the direction of magnetic field will be in 
the direction of stretched thumb. 
(4) Right hand palm rule 
If we stretch our right hand such that fingers point towards the point. At 
which magnetic field is required while thumb is in the direction of current then 
normal to the palm will show the direction of magnetic field. 
 
Note : ? If magnetic field is directed perpendicular and into the plane of the paper it is represented by 
? (cross) while if magnetic field is directed perpendicular and out of the plane of the paper it is 
represented by ? (dot) 
 
 
 
 
In : Magnetic field is away from the observer or perpendicular inwards. 
Out : Magnetic field is towards the observer or perpendicular outwards. 
 Application of Biot-Savarts Law. 
(1) Magnetic field due to a circular current 
If a coil of radius r, carrying current i then magnetic field on it's axis at a distance x from its centre given 
by  
  
3/2 2 2
2
0
) (
2
4 r x
p N i r
.
µ
B
axis
?
?
p
;   where N = number of turns in coil. 
Different cases   
Case 1 :  Magnetic field at the centre of the coil  
(i) At centre x = 0  ? 
r
Ni
B
centre
?
?
? 2
.
4
0
? = 
max
0
2
B
r
Ni
?
?
 
(ii) For single turn coil  N = 1   ? 
r
i
r
i
B
centre
2
2
.
4
0 0
? ?
?
?
? ?      (iii) In C.G.S.    1
4
0
?
?
?
    ? 
r
i
B
centre
? 2
? 
Note : ? N B
centre
?  (i, r  constant),   i B
centre
?   (N, r   constant),  
r
B
centre
1
?    (N, i  constant) 
Case 2 : Ratio of B centre and B axis  
The ratio of magnetic field at the centre of circular coil and on it's axis is given by 
3/2
2
2
1
?
?
?
?
?
?
?
?
? ?
r
x
B
B
axis
centre
 
(i) If 
a c
B B a x 2 2 , ? ? ?          
a c
B B
a
x
8
5 5
,
2
? ? ?        
a c
B B
a
x
2 / 3
2
3
,
2
?
?
?
?
?
?
? ? ? 
(ii) If 
n
B
B
c
a
? then ) 1 (
3 / 2
? ? ?? n r x and if 
n
B
B
c
a
? then ) 1 (
3 / 1
? ? ?? n r x 
Case 3 :  Magnetic field at very large/very small distance from the centre  
i 
B B 
Out In 
i 
B B 
Out In 
i 
CW 
In  
i 
ACW 
Out  
B 
P 
x 
r 
i 
O 
B 
Page 4


 
 
Magnetic Effect of Current 1 
1
genius   PHYSICS 
 
 
 
 
 
 
Oersted found that a magnetic field is established around a current 
carrying conductor. 
Magnetic field exists as long as there is current in the wire. 
The direction of magnetic field was found to be changed when direction of 
current was reversed. 
Note : ? A moving charge produces magnetic as well as electric field, 
unlike a stationary charge which only produces electric field.  
 Biot Savart's Law. 
Biot-Savart’s law is used to determine the magnetic field at any point due to a current carrying 
conductors. 
This law is although for infinitesimally small conductors yet it can be used for long conductors. In order to 
understand the Biot-Savart’s law, we need to understand the term current-element. 
 
Current element  
It is the product of current and length of infinitesimal segment of current carrying wire. 
The current element is taken as a vector quantity. Its direction is same as the direction of current. 
Current element AB = dl i 
 
In the figure shown below, there is a segment of current carrying wire and P is a point where magnetic 
field is to be calculated. l d i is a current element and r is the distance of the point ‘P’ with respect to the current 
element l d i . According to Biot-Savart Law, magnetic field at point ‘P’ due to the current element l d i is given 
by the expression, 
2
r
dlsin ? i
k dB ? also 
? ?
? ?
2
0
sin
.
4 r
dl i
dB B
?
?
?
 
In C.G.S. : k = 1 ? 
2
sin
r
idl
dB
?
? Gauss  
In S.I. : 
?
?
4
0
? k ? 
2
0
sin
4 r
idl
dB
?
?
?
? ? Tesla  
where 
0
? = Absolute permeability of air or vacuum 
metre Amp
Wb
?
? ?
?7
10 4 ? . It's other units are 
metre
Henry
  
or 
2
Amp
N
 or 
Ampere
metre Tesla ?
 
 
 
dl 
i 
P 
r 
A 
B 
dl 
i 
Magnetic lines 
of forces 
i 
 
 
2 Magnetic Effect of Current 
genius   PHYSICS 
 
(1) Different forms of Biot-Savarts law  
Vector form Biot-Savarts law in terms of 
current density 
Biot-savarts law in terms of 
charge and it's velocity 
Vectorially, 
3
0
2
0
) (
4
)
ˆ
(
4 r
r l d i
r
r l d i
B d
?
? ?
?
? ?
? ?
?
?
?
?
?
 ? 
Direction of B d
?
 is perpendicular to 
both l d
?
 and r
ˆ . This is given by 
right hand screw rule. 
 
In terms of current density 
dV
r
r J
B d
3
0
4
?
?
?
?
?
?
?
 
where 
dV
idl
Adl
idl
A
i
j ? ? ? = current 
density at any point of the element, 
dV = volume of element 
In terms of charge and it’s velocity, 
3
0
) (
4
r
r v
q B d
? ?
?
?
?
?
?
 
? v q
dt
l d
q l d
dt
q
l id
?
?
? ?
? ? ? 
 
(2) Similarities and differences between Biot-Savart law and Coulomb’s Law 
(i) The current element produces a magnetic field, whereas a point charge produces an electric field. 
(ii) The magnitude of magnetic field varies as the inverse square of the distance from the current element, 
as does the electric field due to a point charge. 
  
2
0
ˆ
4 r
r l id
B d
?
?
?
?
 Biot-Savart Law r
r
q q
F ˆ
4
1
2
2 1
0
??
? Coulomb’s Law 
(iii) The electric field created by a point charge is radial, but the magnetic field created by a current 
element is perpendicular to both the length element l d and the unit vector r ˆ . 
 
 
 
 
 Direction of Magnetic Field. 
The direction of magnetic field is determined with the help of the following simple laws : 
(1) Maxwell’s cork screw rule 
According to this rule, if we imagine a right handed screw placed along the current 
carrying linear conductor, be rotated such that the screw moves in the direction of flow of 
current, then the direction of rotation of the thumb gives the direction of magnetic lines of 
force.  
 
(2) Right hand thumb rule 
According to this rule if a current carrying conductor is held in the right hand such 
that the thumb of the hand represents the direction of current flow, then the direction of 
folding fingers will represent the direction of magnetic lines of force. 
 
(3) Right hand thumb rule of circular currents 
According to this rule if the direction of current in circular 
B 
i 
B 
dl 
i 
r
ˆ 
B 
r ˆ
 +q 
E 
 
 
Magnetic Effect of Current 3 
3
genius   PHYSICS 
conducting coil is in the direction of folding fingers of right hand, then the direction of magnetic field will be in 
the direction of stretched thumb. 
(4) Right hand palm rule 
If we stretch our right hand such that fingers point towards the point. At 
which magnetic field is required while thumb is in the direction of current then 
normal to the palm will show the direction of magnetic field. 
 
Note : ? If magnetic field is directed perpendicular and into the plane of the paper it is represented by 
? (cross) while if magnetic field is directed perpendicular and out of the plane of the paper it is 
represented by ? (dot) 
 
 
 
 
In : Magnetic field is away from the observer or perpendicular inwards. 
Out : Magnetic field is towards the observer or perpendicular outwards. 
 Application of Biot-Savarts Law. 
(1) Magnetic field due to a circular current 
If a coil of radius r, carrying current i then magnetic field on it's axis at a distance x from its centre given 
by  
  
3/2 2 2
2
0
) (
2
4 r x
p N i r
.
µ
B
axis
?
?
p
;   where N = number of turns in coil. 
Different cases   
Case 1 :  Magnetic field at the centre of the coil  
(i) At centre x = 0  ? 
r
Ni
B
centre
?
?
? 2
.
4
0
? = 
max
0
2
B
r
Ni
?
?
 
(ii) For single turn coil  N = 1   ? 
r
i
r
i
B
centre
2
2
.
4
0 0
? ?
?
?
? ?      (iii) In C.G.S.    1
4
0
?
?
?
    ? 
r
i
B
centre
? 2
? 
Note : ? N B
centre
?  (i, r  constant),   i B
centre
?   (N, r   constant),  
r
B
centre
1
?    (N, i  constant) 
Case 2 : Ratio of B centre and B axis  
The ratio of magnetic field at the centre of circular coil and on it's axis is given by 
3/2
2
2
1
?
?
?
?
?
?
?
?
? ?
r
x
B
B
axis
centre
 
(i) If 
a c
B B a x 2 2 , ? ? ?          
a c
B B
a
x
8
5 5
,
2
? ? ?        
a c
B B
a
x
2 / 3
2
3
,
2
?
?
?
?
?
?
? ? ? 
(ii) If 
n
B
B
c
a
? then ) 1 (
3 / 2
? ? ?? n r x and if 
n
B
B
c
a
? then ) 1 (
3 / 1
? ? ?? n r x 
Case 3 :  Magnetic field at very large/very small distance from the centre  
i 
B B 
Out In 
i 
B B 
Out In 
i 
CW 
In  
i 
ACW 
Out  
B 
P 
x 
r 
i 
O 
B 
 
 
4 Magnetic Effect of Current 
genius   PHYSICS 
(i) If x >> r (very large distance) ? 
3
0
3
2
0
2
.
4
2
.
4 x
NiA
x
Nir
B
axis
?
? ?
?
?
? ? where A = ?r
2
 = Area of each turn of the coil.  
(ii) If x << r (very small distance) ? 
centre axis
B B ? , but by using binomial theorem and neglecting higher 
power of ;
2
2
r
x
 
?
?
?
?
?
?
?
?
? ?
2
2
2
3
1
r
x
B B
centre axis
 
Case 4 : B-x curve  
The variation of magnetic field due to a circular coil as the distance x varies as shown in the figure. 
B varies non-linearly with distance x as shown in figure and is maximum when 0 min
2
? ? x , i.e., the 
point is at the centre of the coil and it is zero at x = ? ?.  
Point of inflection (A and A ?) : Also known as points of curvature change or pints of zero curvature.  
(i) At these points B varies linearly with x ? ?
dx
dB
 constant ? 
0
2
2
?
dx
B d
. 
(ii) They locates at 
2
r
x ? ? from the centre of the coil. 
(iii) Separation between point of inflextion is equal to radius of coil (r)  
(iv) Application of points of inflextion is "Hamholtz coils" arrangement. 
Note : ? The magnetic field at 
2
r
x ? is 
r
Ni
B
5 5
4
0
?
? 
(2) Helmholtz coils 
(i) This is the set-up of two coaxial coils of same radius such that distance between their centres is equal to their 
radius.  
(ii) These coils are used to obtain uniform magnetic field of short range which is obtained between the 
coils. 
(iii) At axial mid point O, magnetic field is given by B
R
Ni
R
Ni
B 432 . 1 716 . 0
5 5
8
0 0
? ? ?
? ?
, where 
R
Ni
B
2
0
?
? 
(iv) Current direction is same in both coils otherwise this arrangement is not called Helmholtz's coil 
arrangement. 
(v) Number of points of inflextion ? Three (A, A ?, A ? ?)  
 
 
 
 
 
 
Note : ? The device whose working principle based on this arrangement and in which uniform 
magnetic field is used called as "Halmholtz galvanometer". 
(3) Magnetic field due to current carrying circular arc : Magnetic field at centre O  
A ? A 
B0 
x = – r/2 
x = 0 x = r/2 
a 
O2 
O 
+ 
– 
a 
O1 
+ 
– 
O 
x 
O2 O1 
2
a
x ? ? 
2
a
x ? 
Resultant field (Uniform) 
A ? 
A ? ? A 
Page 5


 
 
Magnetic Effect of Current 1 
1
genius   PHYSICS 
 
 
 
 
 
 
Oersted found that a magnetic field is established around a current 
carrying conductor. 
Magnetic field exists as long as there is current in the wire. 
The direction of magnetic field was found to be changed when direction of 
current was reversed. 
Note : ? A moving charge produces magnetic as well as electric field, 
unlike a stationary charge which only produces electric field.  
 Biot Savart's Law. 
Biot-Savart’s law is used to determine the magnetic field at any point due to a current carrying 
conductors. 
This law is although for infinitesimally small conductors yet it can be used for long conductors. In order to 
understand the Biot-Savart’s law, we need to understand the term current-element. 
 
Current element  
It is the product of current and length of infinitesimal segment of current carrying wire. 
The current element is taken as a vector quantity. Its direction is same as the direction of current. 
Current element AB = dl i 
 
In the figure shown below, there is a segment of current carrying wire and P is a point where magnetic 
field is to be calculated. l d i is a current element and r is the distance of the point ‘P’ with respect to the current 
element l d i . According to Biot-Savart Law, magnetic field at point ‘P’ due to the current element l d i is given 
by the expression, 
2
r
dlsin ? i
k dB ? also 
? ?
? ?
2
0
sin
.
4 r
dl i
dB B
?
?
?
 
In C.G.S. : k = 1 ? 
2
sin
r
idl
dB
?
? Gauss  
In S.I. : 
?
?
4
0
? k ? 
2
0
sin
4 r
idl
dB
?
?
?
? ? Tesla  
where 
0
? = Absolute permeability of air or vacuum 
metre Amp
Wb
?
? ?
?7
10 4 ? . It's other units are 
metre
Henry
  
or 
2
Amp
N
 or 
Ampere
metre Tesla ?
 
 
 
dl 
i 
P 
r 
A 
B 
dl 
i 
Magnetic lines 
of forces 
i 
 
 
2 Magnetic Effect of Current 
genius   PHYSICS 
 
(1) Different forms of Biot-Savarts law  
Vector form Biot-Savarts law in terms of 
current density 
Biot-savarts law in terms of 
charge and it's velocity 
Vectorially, 
3
0
2
0
) (
4
)
ˆ
(
4 r
r l d i
r
r l d i
B d
?
? ?
?
? ?
? ?
?
?
?
?
?
 ? 
Direction of B d
?
 is perpendicular to 
both l d
?
 and r
ˆ . This is given by 
right hand screw rule. 
 
In terms of current density 
dV
r
r J
B d
3
0
4
?
?
?
?
?
?
?
 
where 
dV
idl
Adl
idl
A
i
j ? ? ? = current 
density at any point of the element, 
dV = volume of element 
In terms of charge and it’s velocity, 
3
0
) (
4
r
r v
q B d
? ?
?
?
?
?
?
 
? v q
dt
l d
q l d
dt
q
l id
?
?
? ?
? ? ? 
 
(2) Similarities and differences between Biot-Savart law and Coulomb’s Law 
(i) The current element produces a magnetic field, whereas a point charge produces an electric field. 
(ii) The magnitude of magnetic field varies as the inverse square of the distance from the current element, 
as does the electric field due to a point charge. 
  
2
0
ˆ
4 r
r l id
B d
?
?
?
?
 Biot-Savart Law r
r
q q
F ˆ
4
1
2
2 1
0
??
? Coulomb’s Law 
(iii) The electric field created by a point charge is radial, but the magnetic field created by a current 
element is perpendicular to both the length element l d and the unit vector r ˆ . 
 
 
 
 
 Direction of Magnetic Field. 
The direction of magnetic field is determined with the help of the following simple laws : 
(1) Maxwell’s cork screw rule 
According to this rule, if we imagine a right handed screw placed along the current 
carrying linear conductor, be rotated such that the screw moves in the direction of flow of 
current, then the direction of rotation of the thumb gives the direction of magnetic lines of 
force.  
 
(2) Right hand thumb rule 
According to this rule if a current carrying conductor is held in the right hand such 
that the thumb of the hand represents the direction of current flow, then the direction of 
folding fingers will represent the direction of magnetic lines of force. 
 
(3) Right hand thumb rule of circular currents 
According to this rule if the direction of current in circular 
B 
i 
B 
dl 
i 
r
ˆ 
B 
r ˆ
 +q 
E 
 
 
Magnetic Effect of Current 3 
3
genius   PHYSICS 
conducting coil is in the direction of folding fingers of right hand, then the direction of magnetic field will be in 
the direction of stretched thumb. 
(4) Right hand palm rule 
If we stretch our right hand such that fingers point towards the point. At 
which magnetic field is required while thumb is in the direction of current then 
normal to the palm will show the direction of magnetic field. 
 
Note : ? If magnetic field is directed perpendicular and into the plane of the paper it is represented by 
? (cross) while if magnetic field is directed perpendicular and out of the plane of the paper it is 
represented by ? (dot) 
 
 
 
 
In : Magnetic field is away from the observer or perpendicular inwards. 
Out : Magnetic field is towards the observer or perpendicular outwards. 
 Application of Biot-Savarts Law. 
(1) Magnetic field due to a circular current 
If a coil of radius r, carrying current i then magnetic field on it's axis at a distance x from its centre given 
by  
  
3/2 2 2
2
0
) (
2
4 r x
p N i r
.
µ
B
axis
?
?
p
;   where N = number of turns in coil. 
Different cases   
Case 1 :  Magnetic field at the centre of the coil  
(i) At centre x = 0  ? 
r
Ni
B
centre
?
?
? 2
.
4
0
? = 
max
0
2
B
r
Ni
?
?
 
(ii) For single turn coil  N = 1   ? 
r
i
r
i
B
centre
2
2
.
4
0 0
? ?
?
?
? ?      (iii) In C.G.S.    1
4
0
?
?
?
    ? 
r
i
B
centre
? 2
? 
Note : ? N B
centre
?  (i, r  constant),   i B
centre
?   (N, r   constant),  
r
B
centre
1
?    (N, i  constant) 
Case 2 : Ratio of B centre and B axis  
The ratio of magnetic field at the centre of circular coil and on it's axis is given by 
3/2
2
2
1
?
?
?
?
?
?
?
?
? ?
r
x
B
B
axis
centre
 
(i) If 
a c
B B a x 2 2 , ? ? ?          
a c
B B
a
x
8
5 5
,
2
? ? ?        
a c
B B
a
x
2 / 3
2
3
,
2
?
?
?
?
?
?
? ? ? 
(ii) If 
n
B
B
c
a
? then ) 1 (
3 / 2
? ? ?? n r x and if 
n
B
B
c
a
? then ) 1 (
3 / 1
? ? ?? n r x 
Case 3 :  Magnetic field at very large/very small distance from the centre  
i 
B B 
Out In 
i 
B B 
Out In 
i 
CW 
In  
i 
ACW 
Out  
B 
P 
x 
r 
i 
O 
B 
 
 
4 Magnetic Effect of Current 
genius   PHYSICS 
(i) If x >> r (very large distance) ? 
3
0
3
2
0
2
.
4
2
.
4 x
NiA
x
Nir
B
axis
?
? ?
?
?
? ? where A = ?r
2
 = Area of each turn of the coil.  
(ii) If x << r (very small distance) ? 
centre axis
B B ? , but by using binomial theorem and neglecting higher 
power of ;
2
2
r
x
 
?
?
?
?
?
?
?
?
? ?
2
2
2
3
1
r
x
B B
centre axis
 
Case 4 : B-x curve  
The variation of magnetic field due to a circular coil as the distance x varies as shown in the figure. 
B varies non-linearly with distance x as shown in figure and is maximum when 0 min
2
? ? x , i.e., the 
point is at the centre of the coil and it is zero at x = ? ?.  
Point of inflection (A and A ?) : Also known as points of curvature change or pints of zero curvature.  
(i) At these points B varies linearly with x ? ?
dx
dB
 constant ? 
0
2
2
?
dx
B d
. 
(ii) They locates at 
2
r
x ? ? from the centre of the coil. 
(iii) Separation between point of inflextion is equal to radius of coil (r)  
(iv) Application of points of inflextion is "Hamholtz coils" arrangement. 
Note : ? The magnetic field at 
2
r
x ? is 
r
Ni
B
5 5
4
0
?
? 
(2) Helmholtz coils 
(i) This is the set-up of two coaxial coils of same radius such that distance between their centres is equal to their 
radius.  
(ii) These coils are used to obtain uniform magnetic field of short range which is obtained between the 
coils. 
(iii) At axial mid point O, magnetic field is given by B
R
Ni
R
Ni
B 432 . 1 716 . 0
5 5
8
0 0
? ? ?
? ?
, where 
R
Ni
B
2
0
?
? 
(iv) Current direction is same in both coils otherwise this arrangement is not called Helmholtz's coil 
arrangement. 
(v) Number of points of inflextion ? Three (A, A ?, A ? ?)  
 
 
 
 
 
 
Note : ? The device whose working principle based on this arrangement and in which uniform 
magnetic field is used called as "Halmholtz galvanometer". 
(3) Magnetic field due to current carrying circular arc : Magnetic field at centre O  
A ? A 
B0 
x = – r/2 
x = 0 x = r/2 
a 
O2 
O 
+ 
– 
a 
O1 
+ 
– 
O 
x 
O2 O1 
2
a
x ? ? 
2
a
x ? 
Resultant field (Uniform) 
A ? 
A ? ? A 
 
 
Magnetic Effect of Current 5 
5
genius   PHYSICS 
 
 
 
 
 
  
r
i
r
i
B
4
.
4
0 0
? ?
?
?
? ?   
r
i
B
?
?
?
.
4
0
?   
r
i
B
) 2 (
.
4
0
? ?
?
? ?
? 
Special results 
 
Angle at centre Magnetic field at 
centre in term of B 0 
360
o
 (2 ?) B 0 
180
o
 ( ?) B 0 / 2 
120
o
 (2 ?/3) B 0 / 3 
90
o
 ( ?/2) B 0 / 4 
60
o
 ( ?/3) B 0 / 6 
30
o
 ( ?/6) B 0 / 12 
 
(4) Concentric circular loops (N = 1)  
(i) Coplanar and concentric : It means both coils are in same plane with common centre 
(a) Current in same direction       (b) Current in opposite direction 
  
                   
?
?
?
?
?
?
?
?
? ?
2 1
0
1
1 1
2
4 r r
i B ?
?
?
                      
?
?
?
?
?
?
? ?
2 1
0
2
1 1
2
4 r r
i B ?
?
?
 
 
 
Note : ?  
?
?
?
?
?
?
?
?
?
?
?
1 2
1 2
2
1
r r
r r
B
B
 
(ii) Non-coplanar and concentric : Plane of both coils are perpendicular to each other  
 
 Magnetic field at common centre  
 
2
2
2
1
0 2
2
2
1
2
i i
r
B B B ? ? ? ?
?
 
 
(5) Magnetic field due to a straight current carrying wire  
Magnetic field due to a current carrying wire at a point P which lies at a perpendicular distance r from the 
wire as shown is given as  
) sin (sin .
4
2 1
0
? ?
?
?
? ?
r
i
B 
From figure ) 90 (
1
? ? ? ?
o
 and  ) 90 (
2
? ? ? ?
o
 
Hence ) cos (cos .
4
? ?
?
?
? ?
r
i
B
o
 
 
 
O r 
i 
O 
i 
r 
? 
i 
r 
? 
O 
i 
i 
r1 
r2 
i 
i 
r1 
r2 
i2 
i1 
B2 
B1 
? 
? 
If magnetic field at the centre of circular 
coil is denoted by B 0 ?
?
?
?
?
?
?
r
i ?
?
? 2
.
4
0
  
Magnetic field at the centre of arc which is 
making an angle ? at the centre is 
?
?
.
2
0
?
?
?
?
?
?
?
B
B
arc
 
P 
?2 
?1 
i 
r 
Y 
X 
? 
? 
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