The figure below shows a circular conducting loop with a 4.70-cm radius and a total resistance of 1.10 Ω placed within a uniform magnetic field pointing into the page.
(a) What is the rate at which the magnetic field is changing if a counterclockwise current
I = 4.70 ✕ 10^−2 A
is induced in the loop? (Give the magnitude.)
(b) Is the induced current caused by an increase or a decrease in the magnetic field with time?
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To find the rate at which the magnetic field is changing, we can use Faraday's law of electromagnetic induction, which states that the induced electromotive force (EMF) in a closed loop is equal to the negative rate of change of magnetic flux through the loop. Mathematically, it can be expressed as:
EMF = -dΦ/dt (Equation 1)
where EMF is the induced electromotive force, dΦ is the change in magnetic flux, and dt is the change in time.
In this case, the induced current in the circular conducting loop is given as I = 4.70 ✕ 10^−2 A.
The magnetic field is uniform and pointing into the page. To determine the magnetic flux, we need to consider the area enclosed by the loop and the magnetic field lines passing through it.
The area of a circle is given by A = πr^2, where r is the radius of the circle. In this case, the radius of the loop is 4.70 cm, so the area of the loop can be calculated as:
A = π(0.0470 m)^2 = 0.00694 m^2 (Equation 2)
Since the magnetic field is uniform, the magnetic flux passing through the loop is given by:
Φ = B * A (Equation 3)
where B is the magnetic field strength.
Substituting Equation 2 into Equation 3, we have:
Φ = B * 0.00694 m^2
Now we can calculate the rate at which the magnetic field is changing by substituting the given values into Equation 1:
EMF = -dΦ/dt
I = -d(Φ)/dt
Rearranging the equation, we find:
d(Φ)/dt = -I
Substituting the given value for I, we have:
d(Φ)/dt = -(4.70 ✕ 10^-2 A)
Therefore, the rate at which the magnetic field is changing is 4.70 ✕ 10^-2 A.
To determine if the induced current is caused by an increase or decrease in the magnetic field with time, we need to analyze Lenz's law. Lenz's law states that the direction of the induced current is such that it opposes the change in the magnetic field that produces it.
Since the induced current in the loop is counterclockwise, it implies that the induced current is trying to create a magnetic field in the opposite direction to the existing field that points into the page.
According to Lenz's law, the induced current is caused by a decrease in the magnetic field with time.