A uniform magnetic field passes through a horizontal circular wire loop at an angle 15.1∘ from the vertical. The magnitude of the magnetic field B changes in time according to the equation B(t)=(3.75 T)+(2.75 Ts)t+(−7.05 Ts2)t2 If the radius of the wire loop is 0.210 m, find the magnitude of the induced emf in the loop when t=5.63 s.
Explanation:
Given that,
A uniform magnetic field passes through a horizontal circular wire loop at an angle 15.1∘ from the vertical, [tex]\theta=15.1^{\circ}[/tex]
The magnitude of the magnetic field B changes in time according to the equation :
[tex]B(t)=3.75+2.75 t-7.05 t^2[/tex]
Radius of the loop, r = 0.21 m
We need to find the magnitude of the induced emf in the loop when t=5.63 s. The induced emf is given by :
[tex]\epsilon=\dfrac{-d\phi}{dt}\\\\\epsilon=\dfrac{-d(BA\cos \theta)}{dt}[/tex]
B is magnetic field
A is area of cross section
[tex]\epsilon=A\dfrac{-dB}{dt}\\\\\epsilon=\pi r^2\dfrac{-d(3.75+2.75 t-7.05 t^2)}{dt}\times \cos\theta\\\\\epsilon=\pi r^2\times(2.75-14.1t)\times \cos\theta[/tex]
At t = 5.63 seconds,
[tex]\epsilon=-\pi (0.21)^2\times(2.75-14.1(5.63))\times \cos(15.1)\\\\\epsilon=10.25V[/tex]
So, the magnitude of induced emf in the loop when t=5.63 s is 10.25 V.