Answer:
The time is 52.6 sec.
Explanation:
Given that,
Radius = 75 cm
Magnetic field
[tex]B(t)=1.4e^{-(0.057)t}[/tex]
We need to calculate the area
[tex]A= \pir^2[/tex]
Put the value into the formula
[tex]A=\pi\times(75\times10^{-2})^2[/tex]
[tex]A=1.767\ m^2[/tex]
We need to calculate the emf
[tex]\epsilon=-\dfrac{dB}{dt}[/tex]
[tex]\epsilon=-\dfrac{d(BA\cos0^{\circ})}{dt}[/tex]
Put the value into the formula
[tex]\epsilon=-\dfrac{d(1.4e^{-(0.057)t}\times1.767\cos0^{\circ})}{dt}[/tex]
[tex]\epsilon=-1.4\times1.767\cos0\times\dfrac{d(e^{-(0.057)t})}{dt}[/tex]
[tex]\epsilon=-2.474\dfrac{d(e^{-(0.057)t})}{dt}[/tex]
[tex]\epsilon=-2.474\times(-0.057)e^{-0.057t}[/tex]
[tex]\epsilon=0.141018e^{-0.057t}[/tex]
For initial value of emf , t = 0
[tex]\epsilon=0.141018e^{0}[/tex]
[tex]\epsilon=0.141018[/tex]
Now, If the induced emf equal to 5% of its initial value
We need to calculate the emf
[tex]\dfrac{5}{100}\times0.141018=0.141018e^{-{0.057t}}[/tex]
[tex]\dfrac{5}{100}=e^{-0.057t}[/tex]
[tex]-0.057t=ln\dfrac{5}{100}[/tex]
[tex]t=\dfrac{2.9957}{0.057}[/tex]
[tex]t=52.6\ s[/tex]
Hence, The time is 52.6 sec.
