Problem 2: A square loop, length L. on each side, is shot with velocity into a...

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Problem 2: A square loop, length L. on each side, is shot with velocity into a uniform magnetic field B. The field is perpendicular to the plane of the loop (into the page in Fig. 2). The loop has mass m and resistance R, and it enters the field at t=0s. Assume that the loop is moving to the right along the x-axis and that the field begins at x = 0 m. Find an expression for the loop's velocity as a function of time as it enters the magnetic field. OPI where the change of the flux FIG. 2: The scheme for Problem 2 a) The induced emf is & = mis due to the change of the area of the loop inside the magnetic field. At the moment of time shown in the scheme this area is equal to xl. Derive the formula for the induced current and express it in terms of the loop's speed = 4. Is the current flowing clockwise or counterclockwise? b) Express the magnetic force that is acting on the loop in terms of B, L, R, and v. Notice that the net force is the force acting on the right edge of the loop, because the forces acting on its top and bottom edges (their parts inside the region with the field) cancel each other. Is the force in the positive or negative i direction? c) Write down the Newton's second law, ma = F, for the loop (use only the magnetic force and ignore the gravitational force). Show that this equation can be reduced to=dt (where does the minus sign come from?). d) Integrate the left side of this equation with respect to and the right side with respect to t from the initial state (=0,=, when the loop enters the field) to the intermediate state (t, v). Assume that the left edge of the loop has not entered the field by the time t. The result will be the expression for as a function of t. Problem 2: A square loop, length L. on each side, is shot with velocity into a uniform magnetic field B. The field is perpendicular to the plane of the loop (into the page in Fig. 2). The loop has mass m and resistance R, and it enters the field at t=0s. Assume that the loop is moving to the right along the x-axis and that the field begins at x = 0 m. Find an expression for the loop's velocity as a function of time as it enters the magnetic field. OPI where the change of the flux FIG. 2: The scheme for Problem 2 a) The induced emf is & = mis due to the change of the area of the loop inside the magnetic field. At the moment of time shown in the scheme this area is equal to xl. Derive the formula for the induced current and express it in terms of the loop's speed = 4. Is the current flowing clockwise or counterclockwise? b) Express the magnetic force that is acting on the loop in terms of B, L, R, and v. Notice that the net force is the force acting on the right edge of the loop, because the forces acting on its top and bottom edges (their parts inside the region with the field) cancel each other. Is the force in the positive or negative i direction? c) Write down the Newton's second law, ma = F, for the loop (use only the magnetic force and ignore the gravitational force). Show that this equation can be reduced to=dt (where does the minus sign come from?). d) Integrate the left side of this equation with respect to and the right side with respect to t from the initial state (=0,=, when the loop enters the field) to the intermediate state (t, v). Assume that the left edge of the loop has not entered the field by the time t. The result will be the expression for as a function of t.