Question

The next two questions refer to this situation: A rectangular loop with sides of length a= 1.00 cm and b= 2.70 cm is placed near a wire that carries a current that varies as a function of time:i(t)=3.62+1.49t2where the current is in Amperes and the time is in seconds. The distance from the straight wire to the closest side of the loop is d= 0.460 centimeters.

Answer 1

a)
`1.007 * 10^(-7)` V.m , into the page

b)
`-5.39 * 10^(-8)` V, counterclockwise

Step-by-step explanation:

a)
`d\Phi = B.dA\\\\B_r = (\mu_0I)/(2\pi r)\\\\dA = b.dr\\\\\Phi = \int\limits^b_a {B.dA} = \int\limits^(a+d)_d {(\mu_0I)/(2\pi r).b.dr} =(\mu_0I)/(2\pi ).b.ln((a+d)/(d) )\\`

At t = 2.90, I = 3.62 + 1.49 * (2.90)^2 = 16.15 A

`\Phi = (4\pi * 10^(-7)*16.15*0.027)/(2\pi) ln(1.46/0.46) = 1.007 * 10^(-7)` V.m

Direction is into the page.

b)
`emf = -d\Phi/dt = -(\mu_0)/(2\pi ).b.ln((a+d)/(d)).(2.98t)= \\=-(4\pi * 10^(-7)*0.027)/(2\pi) ln(1.46/0.46)*2.98*2.90= -5.39 * 10^(-8) V`

Direction is counterclockwise.