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# Solved Problems-4

Problems-4
Derive an expression for the magnetising force at any point on the axis of a single-turn coil carrying a steady current â€˜Iâ€™:
1. When the coil is in the form of a circle of radius â€˜râ€™
2. When the coil is in the form of a rectangle of sides a and b
3. When it is square with sides of length â€˜2aâ€™
Solution
1. By the Biotâ€“Savart law,
âˆ´
âˆ´

By symmetry, the contribution along  adds up to zero because the radial components produced by pairs of current elements 180Â° apart cancel.

âˆ´ Hr = 0

Hence, the magnetic field is given as,

Circular current-carrying loop

Note: At the centre (h = 0), the field is,
1. Let P be a point at a height h above the plane of the loop.
The magnetic field due to the side AB is given as,

âˆ´
Rectangular current-carrying loop

Similarly, the magnetic field due to the side CD is given as,

Since the flow of current in the two elements is in opposite direction, their cosine components will cancel each other and thus, only the axial components will add together.

Hence, the resultant field due to sides AB and CD is given as,

From the figure, it is seen that,

Putting these values, we get,

Similarly, the magnetic field due to the other two sides BC and DA is given as,

Hence, total magnetic field due to the rectangular loop is given as,

1. Along the axis of the coil there will be only a z-component of magnetic field by symmetry. In order to get the total field, it is only necessary to calculate the z-component of the field generated by one side of the coil and then multiply by four. Consider the right-hand side.
Let,
The position of the element of length  is specified by the vector  where,  The position of the point of observation along the z-axis is specified by the vector .
Squarer current-carrying loop

By the Biotâ€“Savart law,

Since all the x-components will add to zero, we have,

By integration,

As the coil has four sides, this must be multiplied by 4 to get the total magnetic field.
âˆ´
âˆ´

Alternative Method

From the result of Part (b), if a = b = 2a then the result becomes,

Note: At the centre (z = 0),  as obtained. This result can becompared with  for a circular coil.