A loop of wire (resistance = 2.0 mΩ) is positioned as shown with respect to a long wire which carries a current. If d = 1.0 cm, D = 6.0 cm, and L = 1.5 m, what current is induced in the loop at an instant when the current in the wire is increasing at a rate of 100 A/s?
A)34 mA
B)30 mA
C)27 mA
D)38 mA
E)0.50 mA

Question

A loop of wire (resistance = 2.0 mΩ) is positioned as shown with respect to a long wire which carries a current. If d = 1.0 cm, D = 6.0 cm, and L = 1.5 m, what current is induced in the loop at an instant when the current in the wire is increasing at a rate of 100 A/s? 
A)34 mA 
B)30 mA 
C)27 mA 
D)38 mA 
E)0.50 mA

Quizplus · Accepted Answer

The induced emf in the loop can be found using Faraday's law:
$\epsilon=-\frac{d\phi_B}{dt}$, where $\phi_B=BAL$, $B$ is the magnetic field due to the current in the long wire, $A$ is the area of the loop, and $L$ is the length of the wire in the loop.

The magnetic field at the center of the loop due to the long wire is $B=\frac{\mu_0I}{2\pi D}$, where $\mu_0$ is the permeability of free space. Substituting the given values, we get $B=3.54	imes10^{-5}$ T.

The area and length of the loop are $A=\pi d^2/4=7.85	imes10^{-5}$ m$^2$ and $L=1.5$ m, respectively.

The total magnetic flux through the loop is then $\phi_B=BAL=3.30	imes10^{-8}$ Wb.

The induced emf is then $\epsilon=-\frac{d\phi_B}{dt}=-\phi_B\frac{d}{dt}(100	ext{ A/s})=3.30	imes10^{-6}$ V.

Finally, the current induced in the loop is $I=\frac{\epsilon}{R}=1.65	imes10^{-3}$ A $=27$ mA.

A Loop of Wire (Resistance = 2