# Magnetic field of a solenoid and toriod

## (5) Magnetic field of a solenoid

• A solenoid is a long wire wound in a close-packed helix carrying a current I and the length of the solenoid is much greater then its diameter
• Figure below shows a section of a stretched out solenoid in xy and yz plane

• The solenoid magnetic field is the vector sum of the field produced by the individual turns that make up the solenoid
• Magnetic field B is nearly uniform and parallel to the axis of the solenoid at interior points near its center and external field near the center is very small
• Consider a dashed closed path abcd as shown in figure .Let l be the length of side ab of the loop which is parallel to the is of the solenoid
• Let us also consider that sides bc and da of the loop are very-very long so that side cd is very much far away from the solenoid and magnetic field at this side is negligibly small and for simplicity we consider its equal to 0
• At side a magnetic field B is approximately parallel and constant. So for this side
∫B.dl=Bl
• Magnetic field B is perpendicular to sides bc and da ,hence these portions of the loop does not make any contributions to the line integral as B.dl=0 for the side bc and da

• Side cd lies at external points solenoid where B.dl=0 as B=0 or negligibly small outside the solenoid

• Hence sum around the entire closed path reduces to Bl
• If N are number of turns per unit length in a solenoid then number of turns in length l is nl.The total current through the rectangle abcd is NIl and from ampere 's law

Bl=μ0NlI
or B=μ0NI                    (22)
• we have obtained this relation for infinitely long solenoids considering the field at external points of the solenoid equal to zero.
• However for real solenoids external field is relatively weak rather then equal to zero
• Thus for actual solenoids relation 22 holds for internal points near the center of the solenoid
• Field at internal points of the solenoid does not depend on length and diameter of the solenoid and is uniform over the cross-section of a solenoid

## (6) Magnetic Field of a toriod

• We will now apply Ampere circuital law to calculate magnetic field of a toriod
• A toriodal solenoid is a hollow circular ring with a large number of turns of a wire carrying current wound around the ring
• Suppose we have to find the magnetic field B at a point P inside the toriod as shown below in figure

• In this case amperion loop would be a circle through point P and concentric inside the toriod
• By symmetry field will have equal magnitude at all points of this circle and this field is tangential to every point in the circle
Thus

• If there are total N number of turns ,net current crossing the area bounded by the circle is NI where I is the current in the toriod
• using Ampere law

Thus we see that field B varies with r i.e. field B is not uniform over the cross-section of the core because the path l=2πr is longer at the outer side of the section then at the inner side
• Imagine a concentric circle through point P' outside the toriod
• The net current passing through this circular disc is zero ,since the current NI passes in and same current passes out. Thus using Ampere's circuital law, the field B=0 outside the torriod

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