Question

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

Answer 1

a) 43.20V

b) 2.71W/s

c) 40.25s

d) 7.77Nm

Step-by-step explanation:

(a) The emf of a rotating coil with N turns is given by:

`emf=NBA\omega sin(\omega t)`

N: turns

B: magnitude of the magnetic field

A: area

w: angular velocity

the emf max is given by:

`emf_(max)=NBA\omega=(50)(1.80T)(0.200m*0.100m)(24.0rad/s)\\\\emf_(max)=43.20V`

(b) the maximum rate of change of the magnetic flux is given by:

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

(c)
`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`

(d) The torque is given by:

`\tau=NABIsin\theta\\\\NAB\omega=emf_(max)\\\\\tau=(emf_(max))/(\omega)(emf_(max))/(R)\\\\\tau=((43.20V)^2)/((24.0rad/s)(10.0\Omega))=7.77Nm`