Answer :
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:
[tex]\[ EMF = -\frac{d\Phi}{dt} \][/tex]
where [tex]\(\Phi\)[/tex] is the magnetic flux and is given by:
[tex]\[ \Phi = B \cdot A \][/tex]
where [tex]\(B\)[/tex] is the magnetic field and [tex]\(A\)[/tex] is the area of the loop.
Given data:
- The radius of the circular loop [tex]\(r = 10.0 \, \text{cm} = 0.1 \, \text{m}\)[/tex]
- The average EMF [tex]\(\epsilon = 20.0 \, \text{mV} = 0.020 \, \text{V}\)[/tex]
- The time interval [tex]\(\Delta t = 1.00 \, \text{s}\)[/tex]
First, we need to find the area [tex]\(A\)[/tex] of the circular loop:
[tex]\[ A = \pi r^2 \][/tex]
Substituting the radius:
[tex]\[ A = \pi (0.1 \, \text{m})^2 = \pi \cdot 0.01 \, \text{m}^2 = 0.01\pi \, \text{m}^2 \][/tex]
Next, we use Faraday's Law to determine the change in magnetic flux [tex]\(\Delta \Phi\)[/tex]:
[tex]\[ EMF = -\frac{\Delta \Phi}{\Delta t} \][/tex]
We can drop the negative sign since we are interested in the magnitude of the change and solve for [tex]\(\Delta \Phi\)[/tex]:
[tex]\[ \Delta \Phi = EMF \cdot \Delta t = 0.020 \, \text{V} \cdot 1.00 \, \text{s} = 0.020 \, \text{Wb} \, (\text{Weber}) \][/tex]
Finally, we find the change in the magnetic field [tex]\(\Delta B\)[/tex] using the relationship between magnetic flux and magnetic field:
[tex]\[ \Delta \Phi = \Delta B \cdot A \][/tex]
Solving for [tex]\(\Delta B\)[/tex]:
[tex]\[ \Delta B = \frac{\Delta \Phi}{A} = \frac{0.020 \, \text{Wb}}{0.01\pi \, \text{m}^2} = \frac{0.020}{0.01\pi} \, \text{T} = \frac{2}{\pi} \, \text{T} \approx 0.636 \, \text{T} \][/tex]
So, the magnetic field has increased by approximately [tex]\(0.636 \, \text{Tesla}\)[/tex] over the 1.00 s interval.
Faraday's Law is given by:
[tex]\[ EMF = -\frac{d\Phi}{dt} \][/tex]
where [tex]\(\Phi\)[/tex] is the magnetic flux and is given by:
[tex]\[ \Phi = B \cdot A \][/tex]
where [tex]\(B\)[/tex] is the magnetic field and [tex]\(A\)[/tex] is the area of the loop.
Given data:
- The radius of the circular loop [tex]\(r = 10.0 \, \text{cm} = 0.1 \, \text{m}\)[/tex]
- The average EMF [tex]\(\epsilon = 20.0 \, \text{mV} = 0.020 \, \text{V}\)[/tex]
- The time interval [tex]\(\Delta t = 1.00 \, \text{s}\)[/tex]
First, we need to find the area [tex]\(A\)[/tex] of the circular loop:
[tex]\[ A = \pi r^2 \][/tex]
Substituting the radius:
[tex]\[ A = \pi (0.1 \, \text{m})^2 = \pi \cdot 0.01 \, \text{m}^2 = 0.01\pi \, \text{m}^2 \][/tex]
Next, we use Faraday's Law to determine the change in magnetic flux [tex]\(\Delta \Phi\)[/tex]:
[tex]\[ EMF = -\frac{\Delta \Phi}{\Delta t} \][/tex]
We can drop the negative sign since we are interested in the magnitude of the change and solve for [tex]\(\Delta \Phi\)[/tex]:
[tex]\[ \Delta \Phi = EMF \cdot \Delta t = 0.020 \, \text{V} \cdot 1.00 \, \text{s} = 0.020 \, \text{Wb} \, (\text{Weber}) \][/tex]
Finally, we find the change in the magnetic field [tex]\(\Delta B\)[/tex] using the relationship between magnetic flux and magnetic field:
[tex]\[ \Delta \Phi = \Delta B \cdot A \][/tex]
Solving for [tex]\(\Delta B\)[/tex]:
[tex]\[ \Delta B = \frac{\Delta \Phi}{A} = \frac{0.020 \, \text{Wb}}{0.01\pi \, \text{m}^2} = \frac{0.020}{0.01\pi} \, \text{T} = \frac{2}{\pi} \, \text{T} \approx 0.636 \, \text{T} \][/tex]
So, the magnetic field has increased by approximately [tex]\(0.636 \, \text{Tesla}\)[/tex] over the 1.00 s interval.
