1) A solenoid produces a uniform magnetic field along its axis and magnetic field outside it is zero. 2) The magnetic field (B) inside a long solenoid is directly proportional to the number of turns per unit length (n) and current (I). 3) At the edges of a long solenoid, the magnetic field is half the value at the center.

1. Magnetic Field
due to Solenoid

3. Solenoid
Solenoid
Solenoid
A wire closely wrapped in form of a helix.
M.F. is produced along Axis of solenoid.
It is used to produce a long-range uniform Magnetic field.
Outside a long solenoid, Magnetic Field is Zero.

6. N : Number of Turns
l : Length of Solenoid
n : Number of Turns per unit length
𝐧 =
𝐍
l
General Parameters of Solenoid
I
R : Radius of Solenoid coil
𝛍𝐫: Relative permeability of core of solenoid
Note
For a long solenoid length of solenoid is very large as compared to
radius of solenoid i.e. l>>R

7. 𝒂
P Q
R
S
න
𝐏
𝑸
𝐁 ∙ 𝐝Ԧ
l + න
𝐐
𝑹
𝐁 ∙ 𝐝Ԧ
l + න
𝐑
𝑺
𝐁 ∙ 𝐝Ԧ
l + න
𝐒
𝑷
𝐁 ∙ 𝐝Ԧ
l
𝐁
= 𝛍𝐨𝐈𝐞𝐧𝐜
Magnetic Field Inside a Long Solenoid
I
By Ampere’s Law,
ර 𝐁 ∙ 𝐝Ԧ
l = 𝛍𝐨𝐈𝐞𝐧𝐜
𝐁𝐚 + 0 + 0 + 0 = 𝛍𝐨𝐈𝐞𝐧𝐜
n : Number of turns per
unit length

8. 𝒂
P Q
R
S
Magnetic Field Inside a Long Solenoid
I
n : Number of turns per
unit length
Now, if number of turns per unit length is n, then
number of turns in length a is N = (𝐧)(𝐚)
𝐁𝐚 = 𝛍𝐨 𝐧𝐚 𝐈
𝐁 = 𝛍𝐨𝐧𝐈
𝐈𝐞𝐧𝐜 = NI = 𝐧𝐚 𝐈
So, current enclosed in the loop
𝐁
𝐁𝐚 + 0 + 0 + 0 = 𝛍𝐨𝐈𝐞𝐧𝐜

9. M.F. in the central region
M.F. in the central region Bo = μonI
Bo = μonI Bo = μo
N
l
I
Bo = μo
N
l
I
M.F. outside the solenoid
M.F. outside the solenoid B = 0
B = 0
M.F. at the edges/end points
M.F. at the edges/end points B =
μonI
2
B =
μonI
2
Magnetic Field Due to Long Solenoid
For a long solenoid, carrying current I :
For a long solenoid, carrying current I :

10. M.F. in the central region
M.F. in the central region Bo = μonI
Bo = μonI Bo = μo
N
l
I
Bo = μo
N
l
I
M.F. outside the solenoid
M.F. outside the solenoid B = 0
B = 0
M.F. at the edges/end points
M.F. at the edges/end points B =
μonI
2
B =
μonI
2
In the presence of Core,
𝛍𝐨 → 𝛍𝐨𝛍𝐫
B
μr : relative
permeability of
core material.
x
B0
Graph between B and x

11. A solenoid of length 80 cm has 500 turns and current through it is 5 A.
Find B.
Example
Example
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧

12. A long solenoid produces magnetic field B at its center. If solenoid is cut
into two parts and same number of turns are wound on one part in double
layer, find magnetic field at the center of new solenoid.
Example
Example
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧

13. A solenoid of length 80 cm has 500 turns and current through it is 5 A.
Another wire having 400 turns and i = 2A is overlapped over it. If both
currents are in opposite directions, then find B.
Example
Example
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧

14. A
B
(N, l)
(N, l)
C
(2N, l)
Solenoid A, B & C have equal resistance. If solenoid A has magnetic field Bo.
Find M.F. in solenoids B & C.
Example
Example
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧

15. P
𝐁𝐏 =
𝛍𝐨𝐧𝐈
𝟐
𝐜𝐨𝐬 𝛟𝟏 + 𝐜𝐨𝐬 𝛟𝟐
ϕ1 ϕ2
Axis
Magnetic Field Due To Finite Length Solenoid
Magnetic field outside a finite length solenoid is not zero.
Magnetic field outside a finite length solenoid is not zero.
Note

16. 𝛟𝟏 → 𝟎°, 𝛟𝟐 → 𝟎°
B = μonI
cos 0° + cos 0°
2
𝛟𝟏 → 𝟎°, 𝛟𝟐 → 𝟗𝟎°
B = μonI
cos 0° + cos 90°
2
For Long Solenoid
In the middle region
In the end region
I
I
𝐁𝐏 =
𝛍𝐨𝐧𝐈
𝟐
𝐜𝐨𝐬 𝛟𝟏 + 𝐜𝐨𝐬 𝛟𝟐
𝐁𝐏 =
𝛍𝐨𝐧𝐈
𝟐
𝐜𝐨𝐬 𝛟𝟏 + 𝐜𝐨𝐬 𝛟𝟐
𝐁 = 𝛍𝐨𝐧𝐈
𝐁 =
𝛍𝐨𝐧𝐈
𝟐

17. If N = 5000, l = 50 cm & I = 5 A, Find BP.
Example
Example
P
60° 53°
Axis
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧

18. A very long solenoid having ‘n’ turns per unit length is shown in the figure.
Find the magnetic field at point P.
Example
Example
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧
P
Axis
R
R
I

19. Magnetic Field
Due To Toroid

20. Magnetic Field Due To Toroid
Toroid is a solenoid
bent in ring shape

21. Magnetic Field Due To Toroid
Toroid is a solenoid
bent in ring shape

22. Magnetic Field Due To Toroid
Toroid is a solenoid
bent in ring shape
𝐑𝟏
𝐑𝟐
In the empty space surrounded by
Toroid (r < R1)
ර 𝐁 ∙ 𝐝Ԧ
l = 𝛍𝐨𝐈𝐞𝐧𝐜
𝐈𝐞𝐧𝐜 = 0
𝐁 = 0
𝒓
𝐀
POINT - A