(a) Calculate the magnetic flux through the loop in the z -direction at the time

ID: 1673885 • Letter: #

Question

(a) Calculate the magnetic flux through the loop in thez-direction at the times indicated below. In each case,specify the direction by giving a positive value for flux in the+z direction and a negative value for flux in the-z direction.

At t = 1.0 s: Fm = T-m2

At t = 3.0 s: Fm = T-m2

At t = 5.0 s: Fm = T-m2 *
0 OK

At t = 6.5 s: Fm = T-m2

At t = 8.0 s: Fm = T-m2 *
0 OK

At t = 9.5 s: Fm = T-m2

HELP: The definition of magneticflux is formally identical to the definition of electric fluxexcept for the substitution of B forE. However, there is a big difference in the rolemagnetic flux plays in practice. In the electrical case, flux isused primarily in Gauss's law and therefore it is the flux over aclosed surface that is primarily of interest. The magneticanalog of net flux through a closed surface is of little interestsince its always zero. However, the magnetic flux through anopen surface is of immense practical importance since itstime-derivative appears on the right-hand side of Faraday's law ofinduction.

(b) Calculate the induced EMF (electromotive force) inthe loop at the times specified below. In each case specify thesense of the EMF by giving a positive value for a counterclockwiseEMF as viewed in the above figure; specify a clockwise sense bygiving a negative value.

At t = 1.0 s: E = V

At t = 3.0 s: E = V *
0 OK

At t = 5.0 s: E = V

At t = 6.5 s: E = V *
0 OK

At t = 8.0 s: E = V

At t = 9.5 s: E = V *

(c) Suppose now that the plane of the loop is tilted and thenheld fixed at an angle q =20° with respect to the x-y plane. Themagnetic field remains in the z-direction and varies withtime exactly as before.

Calculate the magnitude of the magnetic flux throughthe loop and the magnitude of the induced EMF in this caseat t = 1 s.

|Fm| = T-m2

|E| = V

A circular loop of radius a = 10 cm and N=156 turns is fixed in the x-y plane. A spatiallyuniform magnetic field with only a z-component covers theentire area of the loop. The plot at the right showsBz measured in tesla versus time tmeasured in seconds. The +z direction is OUT of thescreen.

Explanation / Answer

for area of the loop

A= a2 = 0.01 m2 wehave

At t = 1.0 s: Fm = B A= 1 x 0.01 =0.031 T-m2 ( looks good)

At t = 3.0 s: Fm = B A = 2 x0.01= 0.063 T-m2

At t = 5.0 s: Fm = B A = 0 x0.01 = 0T-m2 *

At t = 6.5 s: Fm = B A = -2 x 0.01= -0.063T-m2

At t = 8.0 s: Fm = 0T-m2

At t = 9.5 s: Fm = B A = 2 x0.01= 0.063 T-m2

Since emf = N Fm /t ; N= 156turns

emf = N A ( B /t) =

At t = 1.0 s: E EMF= N A ( B /t) = 156 x 0.01 ( 2 /2 ) = 4.9V ( looks good)

please note that they have accepted downbelow (last calculation )

EMF = N A ( B/t) sin(70) = 156 x 0.01 (2/2) sin(70) =+4.6V

and it is the same except it is notsin(70) but a sin (90)

EMF = N A ( B/t) sin(90) = 156 x 0.01 (2/2) sin(70) =+4.9V

At t = 3.0 s: E = B /t=0 thenEMF=0 0 V *

At t = 5.0 s: E = EMF= N A ( B/t) = 156 x 0.01 (-4/2) = -9.8V

At t = 6.5 s: E = 0 V *

At t = 8.0 s: E = EMF= N A ( B/t) = 156 x 0.01 (4/2) =+9.8V

At t = 9.5 s: E = EMF= N A ( B/t) = 156 x 0.01 (0) = 0V *

Calculate the magnitude of the magnetic flux throughthe loop and the magnitude of the induced EMF in this caseat t = 1 s.

|Fm| = BxA= BA sin(90 - 20) = 1 x 0.01 sin(90 - 20 ) =0.030T-m2

EMF = N A ( B /t) sin(70) = 156x 0.01 (2/2) sin(70) = +4.6V

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(a) Calculate the magnetic flux through the loop in the z -direction at the time

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