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Suppose you have a circular loop in space, of
radius 10.0 cm, through which a magnetic flux
passes perpendicular to the loop. The magnetic
field increases a certain amount in a time of
1.00 s. If you measure the average EMF once
around the loop over the 1.00 s interval, and
obtain a value of 20.0 millivolts (mV), by what
amount (T) has the magnetic field increased
over the 1.00 s interval?
[?] T
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68°F Li

Verified Answer :

GinnyAnswer

To solve this problem, we'll use Faraday's Law of Induction, which states that the electromotive force (EMF) around a closed loop is equal to the negative rate of change of the magnetic flux through the loop.

Faraday's Law is given by:
\(\[ EMF = -\frac{d\Phi}{dt} \]\)
where \(\(\Phi\)\) is the magnetic flux and is given by:
\(\[ \Phi = B \cdot A \]\)
where \(\(B\)\) is the magnetic field and \(\(A\)\) is the area of the loop.

Given data:
- The radius of the circular loop \(\(r = 10.0 \, \text{cm} = 0.1 \, \text{m}\)\)
- The average EMF \(\(\epsilon = 20.0 \, \text{mV} = 0.020 \, \text{V}\)\)
- The time interval \(\(\Delta t = 1.00 \, \text{s}\)\)

First, we need to find the area \(\(A\)\) of the circular loop:
\(\[ A = \pi r^2 \]\)
Substituting the radius:
\(\[ A = \pi (0.1 \, \text{m})^2 = \pi \cdot 0.01 \, \text{m}^2 = 0.01\pi \, \text{m}^2 \]\)

Next, we use Faraday's Law to determine the change in magnetic flux \(\(\Delta \Phi\)\):
\(\[ EMF = -\frac{\Delta \Phi}{\Delta t} \]\)
We can drop the negative sign since we are interested in the magnitude of the change and solve for \(\(\Delta \Phi\)\):
\(\[ \Delta \Phi = EMF \cdot \Delta t = 0.020 \, \text{V} \cdot 1.00 \, \text{s} = 0.020 \, \text{Wb} \, (\text{Weber}) \]\)

Finally, we find the change in the magnetic field \(\(\Delta B\)\) using the relationship between magnetic flux and magnetic field:
\(\[ \Delta \Phi = \Delta B \cdot A \]\)
Solving for \(\(\Delta B\)\):
\(\[ \Delta B = \frac{\Delta \Phi}{A} = \frac{0.020 \, \text{Wb}}{0.01\pi \, \text{m}^2} = \frac{0.020}{0.01\pi} \, \text{T} = \frac{2}{\pi} \, \text{T} \approx 0.636 \, \text{T} \]\)

So, the magnetic field has increased by approximately \(\(0.636 \, \text{Tesla}\)\) over the 1.00 s interval.

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