A rectangular coil with 50 turns of conducting wire and a total resistance of 10.0 Ω initially lies in the yz-plane at time t = 0 and rotates about the y-axis with a constant angular speed of 24.0 rad/s. The coil has a height along the y-direction of 0.200 m and a width along the z-direction of 0.100 m. The coil is in the presence of a uniform magnetic field with a magnitude of 1.80 T that points in the +x-direction.

(a) Calculate the maximum induced emf in the coil.

V

(b) Calculate the maximum rate of change of magnetic flux through the coil.

Wb/s

(c) Calculate the induced emf at t = 0.050 0 s.

V

(d) Calculate the torque exerted by the magnetic field on the coil at the instant when the emf is a maximum.

N

[tex]\frac{d\Phi_B}{dt}=\frac{d(A\cdot B)}{dt}=\frac{d}{dt}(ABcos\omega t)=AB\omega sin(\omega t)\\\\\frac{d\Phi_B}{dt}_{max}=(\pi(0.200*0.100))(1.80T)(24.0rad/s)=2.71\frac{W}{s}[/tex]

(c) [tex]emf(t=0.050s)=(50)(1.80T)(0.200m*0.100m)(24rad/s)sin(24.0rad/s(0.050s))\\\\emf(t=0.050s)=40.26V[/tex]

(d) The torque is given by:

[tex]\tau=NABIsin\theta\\\\NAB\omega=emf_{max}\\\\\tau=\frac{emf_{max}}{\omega}\frac{emf_{max}}{R}\\\\\tau=\frac{(43.20V)^2}{(24.0rad/s)(10.0\Omega)}=7.77Nm[/tex]