Respuesta :
Answer:
after 9.00 s ; the magnitude of the induced emf = 0.000944 V
The induced current will be in the clockwise direction.
Explanation:
To find the circumference of the circle ; we use the expression:
C = 2πr
differentiating the above expression to determine the rate of change of the circumference; we have:
[tex]\frac{dC}{dt} = 2 \pi \frac{dr}{dt}[/tex]
Here ; the rate of change of the circumference is [tex]\frac{dC}{dt}[/tex]
Replacing 12.0 cm/s for [tex]\frac{dC}{dt}[/tex]; we have :
[tex]2 \pi \frac{dr}{dt} = - 12.0 cm/s ------ \ (equation \ 1 )[/tex]
However ;the area of the circle is expressed as:
A = πr²
Also; the rate of change of the area is determined as;
[tex]\frac{dA}{dt} = d \frac{\pi r^2}{dt}[/tex]
[tex]\frac{dA}{dt} = 2 \pi r \frac{dr}{dt} ------ \ (equation \ 2)[/tex]
Here; the rate of change of area is [tex]\frac{dA}{dt}[/tex]
Replacing - 12.0 cm/s for [tex]2 \pi r \frac{dr}{dt}[/tex] in equation (2) ; we have:
[tex]\frac{dA}{dt} =(-12.0 \ cm/s) r ------ \ (equation \ 3)[/tex]
Now; after 9.00 s ; the value of circumference of the loop is decreased by:
[tex]C_f = ((12.0 \ cm/s )( \frac{1 \ m}{10^2 \ cm}))(9 \ s)[/tex]
= 1.08 m
The expression for the circumference of the loop after 9.00 s is:
[tex]C = C_i - C_f[/tex]
Given that :
[tex]C_i ( initial \ circumference) = 170 \ cm = 1.7 \ m[/tex]
C = ( 1.7 - 1.08) m
C = 0.62 m
Recall that :
C = 2πr
0.62 = 2×3.14 × r
r = [tex]\frac{0.62}{2*3.14}[/tex]
r = 0.099 m
Replacing r = 0.099 m into equation (3)
[tex]\frac{dA}{dt} =((-12.0 \ cm/s )( \frac{1 \ m}{10^2 \ cm}))(0.099 \ m)[/tex]
[tex]\frac{dA}{dt} =-0.01188 m^2/s[/tex]
From Faraday's law, Induced emf (ε) is expressed as:
[tex]\epsilon = - \frac{d \phi }{dt}[/tex]
and the magnetic flux is given as:
[tex]\phi = BAcos \theta[/tex]
replacing the value of [tex]\phi[/tex] into above equation; we have:
[tex]\epsilon = - \frac{d \ (BAcos \theta) }{dt}[/tex]
= [tex]- B cos \theta \frac{d(A)}{dt}[/tex]
where θ = 0 ; B = 0.800 T and [tex]\frac{dA}{dt}[/tex] = -0.0118 m²/s
[tex]\epsilon = -(0.800 \ T) (cos 0) (-0.0118 m^2/s)[/tex]
[tex]\epsilon = 0.000944 \ V[/tex]
Therefore; after 9.00 s ; the magnitude of the induced emf = 0.000944 V
b) The magnitude of id directing into the plane; However ; considering Lenz's law ; it states that the changes produced in the field will be opposed by the induced current. Thus ; it is found that the direction of the current will be in clockwise direction.
a. The emf induced in the circular loop at the instant when 9.0 seconds have passed is equal to [tex]9.5 \times 10^{-3}\;Volts[/tex] or 0.0095 Volts.
b. Since we assumed that we are facing the circular loop and that the magnetic field points into the circular loop, the direction of the induced current in the loop would be clockwise in accordance with Lenz's law.
Given the following data:
- Initial circumference = 170 cm to m = 1.7 meter.
- Rate of decrement = 12.0 cm/s
- Magnetic field strength = 0.800 T.
- Time = 9 seconds.
a. To find the emf induced in the circular loop at the instant when 9.0 seconds have passed:
First of all, we would determine the circumference of the circular loop.
Mathematically, the circumference of a circle is given by the formula:
[tex]C = 2\pi r[/tex] ...equation 1.
Where:
- r is the radius of a circle.
- C is the circumference of a circle.
The rate of change of the circumference of the circular loop with respect to time is given by:
[tex]\frac{dC}{dt} =2\pi \frac{dr}{t} = -12[/tex] ...equation 2.
For the area of the circular loop:
[tex]A = \pi r^2[/tex] ...equation 3.
The rate of change of the area of the circular loop with respect to time is given by:
[tex]\frac{dA}{dt} =\frac{d}{t} (\pi r^2)\\\\\frac{dA}{dt} =2\pi r\frac{dr}{t}=-12r[/tex] ....equation 4.
After 9 seconds, the circumference of the circular loop is given by:
[tex]C_f = 12 \times 9\\\\C_f = 108\;cm[/tex]
In meter:
Final circumference = [tex]\frac{108}{100} = 1.08\;m[/tex]
[tex]C = Initial\;circumference - Final\;circumference\\\\C = 1.7 -1.08[/tex]
Circumference = 0.62 meter.
Next, we would calculate the radius of of the circular loop:
[tex]C = 2\pi r\\\\0.62=2\times3.142\times r\\\\r=\frac{0.62}{6.284}[/tex]
Radius, r = 0.099 meter
Substituting the value of r into eqn. 4, we have:
[tex]\frac{dA}{dt} =-12r = -12 \times 0.099\\\\\frac{dA}{dt} = -1.188\;cm\\\\\frac{dA}{dt} = -0.01188\;meter[/tex]
Mathematically, the induced emf in a magnetic field is given by the formula:
[tex]E = -B cos\theta(\frac{dA}{dt} )[/tex]
Substituting the parameters into the formula, we have;
[tex]E= -0.800 \times cos(0) \times (-0.01188)\\\\E= -0.800 \times 1 \times (-0.01188)[/tex]
Induced emf (E) = [tex]9.5 \times 10^{-3}\;Volts[/tex] or 0.0095 Volts.
b. Since we assumed that we are facing the circular loop and that the magnetic field points into the circular loop, the direction of the induced current in the loop would be clockwise in accordance with Lenz's law.
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