Two circular loops of wire each have a radius R. They are oriented parallel to e

Question

Two circular loops of wire each have a radius R. They are oriented parallel to each other as shown in the diagram, with one in the z +R plane and the other in the z =-R plane. The z-axis goes through their centers. The current in each is the same (D, but they go in opposite directions as shown. Starting with the Biot-Savart Law, determine an expression for the magnetic field at the point (0, 0, R/2). Evaluate all 5. 0,0,R2) integrals, and your final expression may only contain I, R, x, y, z, and fundamental constants. Clearly indicate the direction of the magnetic field there.

Explanation / Answer

5. given radius = R

location of hoops = Z = +- R

current in eact hoop = I ( in opposite directions)

now from biot savart's law

magnetic field at the axis of a loop of wire carrying current I at distancez from the center is

B = 2*pi*k*R^2*I/sqroot(z^2 + R^2)^3

this can be derived as under

consider a point on axis of the loop, conisder small part of the loop

dB = k*I*dL*R/(z^2 + R^2)^3/2 ( from biot savarts law)

integrating we get te solution

hence

magneitc field at point z = R/2 is

B = 2*pi*k*R^2*I/sqroot(R^2/4 + R^2)^3 - 2*pi*k*R^2*I/sqroot((R + R/2)^2 + R^2)^3

B = 2*pi*k*R^2*I/sqroot(R^2/4 + R^2)^3 - 2*pi*k*R^2*I/sqroot(9R^2/4 + R^2)^3

B = 2*pi*k*R^2*I/sqroot(5R^2/4)^3 - 2*pi*k*R^2*I/sqroot(13R^2/4)^3

B = 2*pi*k*R^2*I[1/sqroot(5R^2/4)^3 - 1/sqroot(13R^2/4)^3]

B = pi*k*I[1/sqroot(5)^3 - 1/sqroot(13)^3]/4R

B = pi*k*I/58.739169R

the magnetic field is in +z direction

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