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Biot - Savart Law: Magnetic field produced along the axis of the current carrying circular coil

Magnetic field produced along the axis of the current carrying circular coil

Consider a current carrying circular loop of radius R and let I be the current flowing through the wire in the direction as shown in Figure 3.40. The magnetic field at a point P on the axis of the circular coil at a distance *z *from its center of the coil O. It is computed* *by taking two diametrically opposite line elements of the coil each of length at C and D. Let be the vector joining the current element (I ) at C to the point P.

According to Biot-Savart’s law, the magnetic field at P due to the current element *I* is

The magnitude of magnetic field due to current element *I* at C and D are equal because of equal distance from the coil. The magnetic field *d *due to each current element *I* is resolved into two components; dB sin θ along y - direction and dB cos θ along z - direction. Horizontal components of each current element cancels out while the vertical components (dB cos θ * *) alone contribute to total magnetic field* *at the point P.

If we integrate around the loop, *d * sweeps out a cone as shown in Figure 3.40, then the net magnetic field at point P is

using Pythagorous theorem r2= R2+Z2 and integrating line element from 0 to 2πR, we get

Note that the magnetic field points along the direction from the point O to P. Suppose if the current flows in clockwise direction, then magnetic field points in the direction from the point P to O.

EXAMPLE 3.17

What is the magnetic field at the center of the loop shown in figure?

Solution

The magnetic field due to current in the upper hemisphere and lower hemisphere of the circular coil are equal in magnitude but opposite in direction. Hence, the net magnetic field at the center of the loop (at point O) is zero .

Tags : Biot - Savart Law | Physics , 12th Physics : Magnetism and Magnetic Effects of Electric Current

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12th Physics : Magnetism and Magnetic Effects of Electric Current : Magnetic field produced along the axis of the current carrying circular coil | Biot - Savart Law | Physics

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