Question

Aflat, circular, steel loop of radius 75 cm is at rest in a uniform magnetic field. The field is perpendicular to the plane of the loop. The field is changing with time, according to an exponential function: B(t)=(1.4T) e- (0.057sec-1)t. When is the induced emf equal to 5% of its initial value? 170 msec 245 msec 52.6 msec O 39.5 msec

Answer 1

The time is 52.6 sec.

Step-by-step explanation:

Given that,

Radius = 75 cm

Magnetic field

`B(t)=1.4e^(-(0.057)t)`

We need to calculate the area

`A= \pir^2`

Put the value into the formula

`A=\pi*(75*10^(-2))^2`

`A=1.767\ m^2`

We need to calculate the emf

`\epsilon=-(dB)/(dt)`

`\epsilon=-(d(BA\cos0^(\circ)))/(dt)`

Put the value into the formula

`\epsilon=-(d(1.4e^(-(0.057)t)*1.767\cos0^(\circ)))/(dt)`

`\epsilon=-1.4*1.767\cos0*(d(e^(-(0.057)t)))/(dt)`

`\epsilon=-2.474(d(e^(-(0.057)t)))/(dt)`

`\epsilon=-2.474*(-0.057)e^(-0.057t)`

`\epsilon=0.141018e^(-0.057t)`

For initial value of emf , t = 0

`\epsilon=0.141018e^(0)`

`\epsilon=0.141018`

Now, If the induced emf equal to 5% of its initial value

We need to calculate the emf

`(5)/(100)*0.141018=0.141018e^{-{0.057t}}`

`(5)/(100)=e^(-0.057t)`

`-0.057t=ln(5)/(100)`

`t=(2.9957)/(0.057)`

`t=52.6\ s`

Hence, The time is 52.6 sec.