Physics 2420 In-Class Problems: Electromagnetic Induction & Faraday's Law

1. In the diagram below, a circular loop immersed in a magnetic field B. The field is oriented at an angle of θ = 25° to normal to the loop. The radius, r₀, of the loop is 10 cm. Determine the emf and the direction of the current if
   
   1. If B = 0.045 T;
   2. If B = 0.050cos(Ωt);
   3. If B = 0.045 T and r = 0.5r₀t²;
   4. If B = -0.035t²+0.045;
   

2. A single circular hoop moves with constant velocity through regions where uniform magnetic fields of the same magnitude are directed either into or out of the plane of the page as indicated below. Determined the direction of the induced current, if any, at each of the seven marked positions. HINT: sketch the flux as a function of position.
   
   

3. The DC-10 jet aircraft has a wingspan of 47 m. If such an aircraft is flying horizontally at 960 km/h at a place where the vertical component of the earth's magnetic field is 60 μT, what is the induced emf between its wingtips?

4. In diagram (a) below, an equilateral triangle is just entering, at time t = 0, a region of constant magnetic field B = 0.335 T into the page. In diagram (b) at some later time t > 0, the triangle has moved a distance x into the magnetic field. The triangle has sides of length L = 1.20 m long and is moving to the right at constant speed v = dx/dt = 2.50 m/s.
   
   1. Derive an expression for the magnetic flux φ_m as a function of x. Hints: The area of a triangle is one-half the base times the height. Consider similar triangles.
      
   2. What is the magnitude of the induced emf at t = 0.30 s?
      
   3. What is the direction of the induced emf at t = 0.30 s? Fully explain your reasoning.
      
   4. If the resistance of the wire is 0.50 Ω, what is the current in the wire?
      
   

5. Two parallel conducting rails are inclined at 30.0 to the horizontal, and are joined at the top by a length of copper wire; the rails and wire have negligible resistance. A 0.40 m long conducting rod of resistance 2.00 Ω slides without friction down the rails. Sliding through the magnetic field induces a current in the rod. The current-carrying rod then experiences a force from the external magnetic field. Assuming that the component of the magnetic field perpendicular to the incline, B, points up, what magnitude must it have to ensure that the rod slides with a constant velocity of 5.00 m/s. The mass of the rod is 50.0 g. If the perpendicular component of the magnetic field pointed down what effect would this have? Why can we neglect the parallel component of the magnetic field, B?
   
   

6. Determine which of the following electric fields are non-conservative magnetically induced field. Determine the magnetic field which produced the non-conservative field.
   
   1. E_x = x+y, E_y = -x+y, and E_z = -2z,
      
   2. E_x = 2y, E_y = 2x+3z, and E_z = 3y,
      
   3. E_x = x²-z², E_y = 2, and E_z = -2xz,
   4. E_x =(y³-z³)cos(Ωt), E_y = (z³-x³)cos(Ωt), and E_z = (x³-y³)cos(Ωt).
      

Questions? mike.coombes@kwantlen.ca