A wire loop of radius 0.250 m is placed in a uniform magnetic field of magnitude 0.740 T. The field is directed perpendicular to the plane of the loop.
-If the field reverses its direction,and its magnitude changes to 0.100 T in time 1.06 s,what is the magnitude of the average induced emf in the loop during this time?
A)156 mV
B)119 mV
C)185 mV
D)741 mV

Question

A wire loop of radius 0.250 m is placed in a uniform magnetic field of magnitude 0.740 T. The field is directed perpendicular to the plane of the loop. 
-If the field reverses its direction,and its magnitude changes to 0.100 T in time 1.06 s,what is the magnitude of the average induced emf in the loop during this time?
A)156 mV
B)119 mV
C)185 mV
D)741 mV

Quizplus · Accepted Answer

The magnitude of the average induced emf can be calculated using Faraday's law of electromagnetic induction, which states that the induced emf ($\epsilon$) in a loop is equal to the rate of change of magnetic flux ($\Phi_B$) through the loop. Mathematically, this is given by $\epsilon = -\Delta \Phi_B / \Delta t$, where $\Delta \Phi_B$ is the change in magnetic flux and $\Delta t$ is the time over which this change occurs.The magnetic flux $\Phi_B$ through the loop is given by $\Phi_B = B \cdot A \cdot \cos(\theta)$, where $B$ is the magnetic field strength, $A$ is the area of the loop, and $\theta$ is the angle between the magnetic field and the normal to the plane of the loop. Since the field is perpendicular to the loop, $\cos(\theta) = 1$, simplifying the expression to $\Phi_B = B \cdot A$.Given that the radius of the loop is 0.250 m, the area $A$ of the loop is $\pi r^2 = \pi \cdot (0.250)^2$ m$^2$. The change in magnetic field $\Delta B$ is from 0.740 T to -0.100 T (a total change of 0.840 T, since it reverses direction and decreases in magnitude), and the change in time $\Delta t$ is 1.06 s.The change in magnetic flux $\Delta \Phi_B$ is then $\Delta B \cdot A = 0.840 \cdot \pi \cdot (0.250)^2$. Plugging this into the formula for induced emf gives:$$\epsilon = -\frac{\Delta \Phi_B}{\Delta t} = -\frac{0.840 \cdot \pi \cdot (0.250)^2}{1.06}$$Calculating this gives:$$\epsilon = -\frac{0.840 \cdot 3.1416 \cdot 0.0625}{1.06} \approx -0.156 \text{ V} = 156 \text{ mV}$$The negative sign indicates the direction of the induced emf (opposite to the change in flux), but since the question asks for the magnitude, we take the absolute value, which is 156 mV.

A Wire Loop of Radius 0