Homework #8: Magnetic Force and Biot-Savart Law 1. [10 points] Suppose that an electron is released from rest in a region where there is a uniform electric field in the z-direction and a magnetic field in the x-direction. Determine the trajectory of the electron if it starts at the origin with velocity. )̂ a. ⃗( ) ( )̂ b. ⃗( ) ( )( ̂ c. ⃗( ) ( ̂) 2. Consider a rectangular loop of wire, supporting a mass m, which hangs vertically with one end in a uniform magnetic field ⃗⃗, which points into the page. a. [3 points] What current I in the loop would exactly balance the weight of the mass? b. [7 points] Suppose we now increase the current such that the magnetic force exceeds the weight of the mass. This causes the loop to rise and thus, lifts the mass. What is the work done to lift the wire? What force is doing work here? Defend your answer. 3. Consider a circular wire loop of radius R located in the yz plane and carrying a steady current I, as shown below. a. [7 points] Calculate the magnetic field at an axial point P a distance x from the center of the loop. b. [3 points] Show that the magnitude of the magnetic field far from the loop is given by Here, ( ) is defined as the magnetic moment of the loop. 4. Magnetic Force Exerted on Current Carrying Loops a. [5 points] Find the force on a square loop near an infinite straight wire. Assume that both the loop and the wire carry a steady current I. b. [5 points] Find the force on the triangular loop. Assume that the loop and wire carry a steady current I. 5. [10 points] A semicircular wire carries a steady current I as shown below. Show that the magnetic field at a point P on the other semicircle is [ ( ) ( ) ] Bonus [7 points]: Using the Biot-Savart law, show that the magnetic field at a point axis of the rotating sphere, as shown below, is given by ⃗⃗ Here, ̂ is the angular speed of a charge element around the z-axis Hint: The following integrals may be helpful ∫ √ [( ∫ ∫ √ ) ) ] ( √ ( ) on the