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Chapter cover: Magnetic field due to a straight wire, Circular Loop, Circular Arc.
When k is proportionally constant.
Here dl vector points in the direction of current i.
Equation (1) is the vector form of the Biot-Savart Law. The magnitude of the field induction at P is given by
Where Ѳ is the angle between dl and r.
If the medium is other than air or vacuum, the magnetic induction is
Where μr is relative permeability of the medium and is a dimensionless quantity.
FIELD DUE TO A STRAGHT CURRENT CARRYING WIRE
3.1 WHEN THE WIRE IS OF FINITE LENGTH
Consider a straight wire segment carrying a current I and there is a point P at which magnetic field to b e calculated as shown in the figure. This wire segment makes angle Ѳ1 and Ѳ2 at that point with normal OP. Consider an element of length dy at a distance y from O and distance of this elements from points P is r and line joining P to Q makes an angle Ѳ with the direction of current as shown in figure. Using Biot-Savart Law magnetic field at point P due to small current element is given by
As every element of the wire contributes to B in the same direction, we have
From the triangle OPQ as shown in diagram, we have
y=d tan ϕ or dy= d sec2 ϕ d ϕ and is same triangle,
r=d sec ϕ and Ѳ = (900 - ϕ), where ϕ is angle between line OP and PQ
Now e.g. (1) can be written in this form
Ѳ1 and Ѳ2 must be taken with sign
For the case shown in figure
Direction of: The direction of magnetic field is determined by the cross product of the vector idl with r. Therefore, at point P, the direction of the magnetic field due to the whole conductor will be perpendicular to the plane of paper and going into the plane.
Right-hand Thumb Rule: The direction of B at a point P due to a long, straight wire can be found by the right-hand thumb rule. The direction of magnetic field is perpendicular to the plane containing wire and perpendicular from the point. The orientation of magnetic field is given by the direction of curl fingers if we stretch thumb along the wire in the direction of current. Refer figure.
Conventionally, the direction of the field perpendicular to the plane of the paper is represented by ⨂ if into the page and by ⦿ If out of the page. Now consider some special cases involving the application of equation (3)
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