CBSE Class 12-science Answered

State and explain biot savart law ,give it's application to circular loop carrying current at a point on the axis of the circular loop

Asked by ap996969 | 23 Feb, 2019, 09:24: AM

Expert Answer

Figure shows a finite conductor XY carrying current I. Let us consider an infinitesimal element dl of the conductor.

The magnetic field dB due to this element is to be determined at a point P which is at a distance r from it.

Let θ be the angle between dl and the displacement vector r.

Biot-Savart’s law expresses that the magnitude of the magnetic field dB is proportional to the current I, the element length |dl|,
and inversely proportional to the square of the distance r. Its direction is perpendicular to the plane containing dl and r .

Thus, in vector notation,

begin mathsize 16px style stack d B with rightwards arrow on top space proportional to space fraction numerator I space stack d l with rightwards arrow on top space cross times r with rightwards arrow on top over denominator r cubed end fraction end style

and

begin mathsize 16px style stack d B with rightwards arrow on top space equals space fraction numerator mu subscript 0 over denominator 4 straight pi end fraction space fraction numerator I space stack d l with rightwards arrow on top space cross times r with rightwards arrow on top over denominator r cubed end fraction end style

where μ0/4π is a constant of proportionality. The above expression holds when the medium is vacuum.

The magnitude of this field is,

begin mathsize 16px style open vertical bar d B close vertical bar space equals space fraction numerator mu subscript 0 over denominator 4 straight pi end fraction fraction numerator I space d l space sin theta over denominator r squared end fraction end style

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Figure depicts a circular loop carrying a steady current I. The loop is placed in the y-z plane with its centre at the origin O and has a radius R.

The x-axis is the axis of the loop. We wish to calculate the magnetic field at the point P on this axis. Let x be the distance of
P from the centre O of the loop. Consider a conducting element dl of the loop. This is shown in figure.

The magnitude dB of the magnetic field due to dl is given by the Biot-Savart law

Now r² = x² + R² . Further, any element of the loop will be perpendicular to the displacement vector from the element to the axial point.

For example, the element dl in Figure is in the y-z plane whereas the displacement vector r from dl to the axial point P is in

the x-y plane. Hence |dl × r|=r dl. Thus,

begin mathsize 12px style d B space equals space fraction numerator mu subscript 0 over denominator 4 straight pi end fraction fraction numerator I space d l over denominator open parentheses x squared space plus space R squared close parentheses end fraction end style

....................(1)

The direction of dB is shown in Figure. It is perpendicular to the plane formed by dl and r. It has an x-component dB_x and a component
perpendicular to x-axis, dB_⊥. When the components perpendicular to the x-axis are summed over, they cancel out and we obtain a null result.
For example, the dB_⊥ component due to dl is cancelled by the contribution due to the diametrically opposite dl element, shown in figure.
Thus, only the x-component survives. The net contribution along x-direction can be obtained by integrating dB_x = dB cos θ over the
loop. From Figure,