Experiment: Magnetic Induction due to a long Solenoid Carrying Current - QS Study
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Experiment: Magnetic induction due to a long solenoid carrying current

Let consider an infinitely long solenoid having n turns per unit length carrying a current of I. For such an ideal solenoid (whose length is very large compared to its radius), the magnetic field at points outside the solenoid is zero.

A long solenoid appears like a long cylindrical metal sheet (Figure). The upper view of dots is like a uniform current sheet coming out of the plane of the paper. The lower row of crosses is like a uniform current sheet going into the plane of the paper.

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Fig: Magnetic field due to a long solenoid

To find the magnetic induction (B) at a point inside the solenoid, let us consider a rectangular Amperean loop abcd. The line integral ∫ B.dl for the loop abcd is the sum of four integrals.

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If l is the length of the loop, the first integral on the right side is Bl. The second and fourth integrals are equal to zero because B is at right angles for every element dl along the path. The third integral is zero since the magnetic field at points outside the solenoid is zero.

∴ ∫ B.dl = Bl … (1)

Since the path of integration includes nl turns, the net current enclosed by the closed loop is

I0 = Inl … (2)

Ampere’s circuital law for a closed loop is

∫ B.dl = μ0I0 … (3)

Substituting equations (1) and (2) in equation (3)

Bl = μ0 Inl

∴ B = μ0 nI … (4)

The solenoid is commonly used to obtain uniform magnetic field.

By inserting a soft iron core inside the solenoid, a large magnetic field is produced

B = μnI = μ0 μ1 nI … (5)

when a current carrying solenoid is freely suspended, it comes to rest like a suspended bar magnet pointing along north-south.