4. Use Faraday's law. You need to express rate of change of flux in terms of area and magnetic field. Decide which of these is changing and express this rate in some way. Here it is the magnetic field, not the area that is changing; express this as a derivative of the function given for B.
5. Rate of change of flux is derivative of flux function, evaluated at t = 2. (b) field shown is increasing out of the page; from Lenz's law, induced emf will produce opposing field into page.
11. Rate of change of flux depends on changing field in solenoid produced by changing current. Use eq'n for ideal solenoid, express rate of change of current using given information.
19. Once again changing flux depends on changing magnetic field. Rate of change is found from derivative of function. Total emf equals battery and induced emf combined. Find direction of induced emf using Lenz's law: magnetic field out of the page is decreasing, induced emf will produce supporting field.
23. Here change in flux is created by changing area as rod moves. Let x be distance from right-hand end of rails and L be distance between rails. Then flux = BA = Blx and from Faraday's law emf = Bldx/dt = Blv; (b) Ohm's law
24. (a) same as # 23 (0.60 V) (b) use Ohm's law to find magnitude. Since area is increasing, flux increases and induced current must produce opposing field. (1.5 A CW) (c) P = I2R (0.90 W) (d) At constant velocity, net force = 0 so applied force must be equal and opposite magnetic force. (0.18 N to left) (e) P = Fv (0.90 W, same)
28. (a) emf will change direction each half turn. (same) (b) This is basically an ac generator. Rate of change of flux depends on amount of magnetic field lines enclosed by the loop as it turns. It will be a maximum when the loop is in the plane of the page and will vary sinusoidally. The amplitude is the maximum value obtained when the sin of the angle between the field and the normal line of the coil is at its maximum, and equals 1. The flux = BA cos (theta), where theta is the angle between the field and the normal line of the coil. Theta, the angular displacement is equal to the angular frequency (omega) times the time. The emf then equals the derivative, with respect to time, of [ BA cos (omega) t]. (pi2a2Bf)
40. (a) Use eqn 32-21, Faraday's law. Rate of change of flux is rate of change of field times the area enclosed by 2.2 cm circle. In integral on left side of eqn, ds evaluates to circumference of 2.2 cm circle. (7.15 x 10-5 V/m) (b) Outside solenoid, integral of ds is circumference of 8.2 cm circle, while area of magnetic field is simply cross sectional area of solenoid's 6 cm circle. (1.43 x 10-4 V/m)
41. For first two regions, integral = -(area of region) x (rate of change of magnetic field). For third region, since fields are opposite, total integral is difference between first two regions.