Respuesta :
Answer:
The current induced in the loop is [tex]\dfrac{\pi a^2 B\omega}{4R}\ \sin(\omega t)[/tex].
Explanation:
Given that,
Magnetic field [tex]B(t)=B\cos(\omega t)\ z[/tex]
Radius [tex]r =\dfrac{a}{2}[/tex]
Resistance =R
We need to calculate the area of the loop
Using formula of area
[tex]A = \pi r^2[/tex]
Put the value of r in to the formula
[tex]A =\pi\times(\dfrac{a}{2})^2[/tex]
[tex]A=\dfrac{\pi a^2}{4}[/tex]
We need to calculate the flux
Using formula of flux
[tex]\phi=BA[/tex]
[tex]\phi=B\cos(\omega t)\dfrac{\pi a^2}{2}[/tex]
We need to calculate the emf
Using formula of emf
[tex]\epsilon=-\dfrac{d\phi}{dt}[/tex]
[tex]\epsilon=-\dfrac{-\pi a^2B}{4}(-\omega\sin\omega t)[/tex]
[tex]\epsilon=\dfrac{\pi a^2B\omega\sin(\omega t)}{4}[/tex]
We need to calculate the current
Using formula of current
[tex]I(t)=\dfrac{\epsilon}{R}[/tex]
[tex]I(t)=\dfrac{\dfrac{\pi a^2B\omega\sin(\omega t)}{4}}{R}[/tex]
[tex]I(t)=\dfrac{\pi a^2 B\omega}{4R}\ \sin(\omega t)[/tex]
Hence, The current induced in the loop is [tex]\dfrac{\pi a^2 B\omega}{4R}\ \sin(\omega t)[/tex].
