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Example 22.6 Magnetic Field on the Axis of a Circular Current Loop Consider a circular wire loop of radius a located in the yz plane and carrying a steady current I as in the figure below. Calculate the magnetic field at an axial point Pa distance x from the center of the loop SOLVE IT dB dB . Conceptualize The figure shows the magnetic field contribution dB at P due to a single current element at the dB, top of the ring. This field Geometry for calculating the magnetic field at a point Plying on the axis of a vector can be resolved into current loop. By symmetry, the total fald Bls along this axis. components de parallel to the axis of the ring and dB perpendicular to the axis. Think about the magnetic field contributions from a current element at the bottom of the loop. Because of the symmetry of the situation, the perpendicular components of the field due to elements at the top and bottom of the ring cancel. This cancellation occurs for all pairs of segments around the ring, so we can ignore the perpendicular component of the field and focus solely on the parallel components, which simply add. Categorize We are asked to find the magnetic field due to a simple current distribution, so this example is a typical problem for which the Biot-Savart law is appropriate. Analyze In the situation, every length element ds is perpendicular to the vector at the location of the element. Therefore, for any element, lds x 1 = (ds)(1) sin 90° - ds. Furthermore, all length elements around the loop are at the same distance from P, where ?-? + x2 Use the equation to the right to find the Hol ds magnitude of dB due to the current in any 4/1 (+ x2) length element ds: dB - Mollas y Find the x component of the field element: de- Hol ds 4T (at cos

Answer 1

Magnetic field (B)= $\mu$ o I a²/2×{(x²+a²)^3/2}