Question

4. (2 pts) Find the exact magnetic field a distance z above the center of a rectangle lying in the xy plane, centered on the origin, with side of length a in the x direction and side of length b in the y direction, carrying a current. Verify that it reduces to the field of a dipole with the appropriate dipole moment when za and z»b.

Answer 1

Let a rectangular coil according to given information in the question shown in fig. Hence, simply magnetic field at point P is the combination of 4 wires (finite wire). Now As we know magnetic field due to finite wire is given as B_P = mu_0 I/4 pi d (sin theta_1 + sin theta_2) and if angles are some, the B_P = mu_0 I/4 pi d (2 sin theta) Now, Let�s come to the question, B_P = mu_0 I/4 h mu (2 sin theta) Now have we need to calculate mu, sin theta, to calculate B at point P due to AB wire. \r\n calculation of mu From right angled triangle POM. We have mu^2 = b^2/4 + z^2 So, mu = squareroot b^2/4 + z^2 Also using right angled triangle PM B. Sin theta = P/H = a/2/squareroot a^2/4 + mu^2 = a/2/squareroot a^2/4 + b^2/4 + z^2 So putting these values in above eq we will get