b) Calculate the force on each side in x- and y- components. c) Add the components from all sides and show that the net force on the triangle is zero
P118 2018W WS09 Torque on a current-carrying loop Consider the forces on the different sides of a current loop. The plane of the loop is at an angle with respect to the vertical axis. We can calculate direction and magnitude of the magnetic force acting on each side using ? 𝐵 = 𝐼𝐿 ⃗ × ? ⃗ The directions are given by the right-hand rule and we find that the forces on opposite sides are of the same magnitude and point in opposite directions. The forces on the longer sides are exerted around a moment arm leading to a net torque. The forces on the smaller sides are along the same line and cancel. The picture shows these forces, viewed from the side as indicated by the pair of eyes. Q11.17 Show that the net torque on the rectangular loop can be written as 𝝉 ⃗ = 𝝁 ⃗ × 𝑩 ⃗⃗ The magnetic dipole moment is defined as 𝜇 = 𝐼? where ? is the area vector of the loop and the direction is determined by a right- hand rule. I is the current in the loop. Start with the definition of the torque 𝜏 = ? × ? and its magnitude 𝜏 = ?? sin𝜃 and the expressions for the torque:
P118 2018W WS09 Q11.18 If released from rest, the current loop will A. Move upward. B. Move downward. C. Rotate clockwise. D. Rotate counterclockwise. E. Do something not listed here. Draw the directions of the forces on opposite sides of the ring. (The picture shows a cut through the ring.) Q11.19 A current loop, carrying a current of 3.5 A, is in the shape of an equilateral triangle with sides 20 cm. The loop is in a uniform magnetic field of magnitude 100 mT whose direction is parallel to the current of the right side of the loop. a) Find the magnitude of the magnetic dipole moment. b) Calculate the torque on the loop. c) Assuming that the loop is free to rotate in any direction, in which direction will it rotate? Draw the rotation axis in the figure and show the torque as a vector. Answers: 0.07 A m 2 ; 0.007 N m
P118 2018W WS09 Potential Energy of a loop in a uniform magnetic field. The magnetic dipole moment tends to align with the external magnetic field . This applies not only to loops, but also to compass needles and atoms. For example, the magnetic interaction between electrons inside an atom gives rise to spin- orbit and spin-spin interaction that influence their potential energy. Q11.20 In the semi-classical Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius 5.3 x 10 -11 m with a speed of 2.2 x 10 6 m/s. a) What is the magnetic moment due to the moving electron? b) The potential energy of an object with magnetic dipole moment u is given by 𝑈 = −𝜇 ∙ ? ⃗ According to the Bohr model and assuming that a hydrogen atom is initially aligned in direction of a magnetic field of strength 1.0 T, what energy would be needed to flip a hydrogen atom in opposite direction?
- Fall '19