Question

3. (a) A square loop of wire lies in the ry plane and is centred at the origin. Two segments are parallel to the axis and are at y = ±R. The other two segments are parallel to the y axis and are at z-th. The loop carries a steady current lo. What is the magnetic field at the origin? (b) The square loop of part (a) is replaced by a hexagon: the distance from the origin to the nearest point on any of the six sides is R. What is its magnetic field at the origin? (c) The hexagon of part (b) is replaced by a regular N-sided polygon, with the distance from the origin to the nearest point on any of the N sides being R. What is its magnetic field at the origin? You may wish to check that บ。ur answer approaches the magnetic field of a ring of rudus R -Le., your answer to question #1 for the special case = 0-as you let N → .

Answer 1

Magnetic field at the center of a squarer. Magnetic field due to a wire: - B_P = mu_0 I/4 Gd [sin theta_1 + sin theta_2] so magnetic field at the center of a square is that B_P = 4 middot mu_0 I/4G middot (R/2) times J_2 = 2 J_2 mu_0 I/4 G R (downward.) B_P = 2 J_2 mu_0 I/4 GR (downward) \r\n Magnetic field at the center of a Hexagon d = R cos 130 degrees = R times J_3/2 B_P = mu_0 middot I/4G times R/2 J_3 [1/2 + 1/2] times (6) number of wire. B_P = squareroot 3 mu_0 I/GR (downward.) Magnetic field at the center of a polygon. Angle subtend by n side at center is that = 201 n rightarrow 2G 1 rightarrow 2G/h half angle rightarrow G/n d rightarrow a cos(G/n) \r\n Magnetic field at the center is that: - B_center = mu_0 I/4 Gd [sin theta_1 + sin theta_2] B_center = mu_0