A long, narrow rectangular loop of wire is moving toward the bottom of the page with a speed of 0.021 m/s (see the drawing). The loop is leaving a region in which a 2.7 T magnetic field exists; the magnetic field outside this region is zero. During a time of 3.0 s, what is the magnitude of the change in the magnetic flux?

To find the magnitude of the change in magnetic flux, we need to calculate the initial and final magnetic flux and then subtract them.

The formula for magnetic flux is given by:

Φ = B * A * cos(θ)

Where:
Φ is the magnetic flux,
B is the magnetic field strength,
A is the area of the loop, and
θ is the angle between the magnetic field and the normal to the loop.

In this case, the loop is long and narrow, so we can assume that the magnetic field is parallel to the normal to the loop. Therefore, θ = 0° and cos(θ) = 1.

Given that the magnetic field strength inside the region is 2.7 T and the speed of the loop is 0.021 m/s, we can calculate the change in area.

To do this, we can use the equation:

ΔA = v * t

Where:
ΔA is the change in area,
v is the velocity of the loop, and
t is the time interval.

Given that the time interval is 3.0 s, we can calculate the change in area:

ΔA = 0.021 m/s * 3.0 s
ΔA = 0.063 m²

Now, we can calculate the initial and final magnetic flux:

Initial magnetic flux (Φ₁) = B * A₁ * cos(θ)
Final magnetic flux (Φ₂) = B * A₂ * cos(θ)

Since the magnetic field outside the region is zero, the final magnetic flux is zero (Φ₂ = 0). Therefore, we only need to calculate the initial magnetic flux.

Initial magnetic flux (Φ₁) = 2.7 T * A₁ * cos(θ)

Finally, the magnitude of the change in magnetic flux (ΔΦ) is given by:

ΔΦ = Φ₂ - Φ₁
ΔΦ = 0 - (2.7 T * A₁ * cos(θ))

Thus, to find the magnitude of the change in the magnetic flux, we need to know the initial area of the loop (A₁) and the angle (θ) between the magnetic field and the normal to the loop.

To calculate the magnitude of the change in the magnetic flux, we can use Faraday's Law of Electromagnetic Induction. According to this law, the magnitude of the change in the magnetic flux is equal to the induced electromotive force (emf) in the loop, divided by the rate of change of time.

The equation for Faraday's Law is:

ε = -N * ΔΦ/Δt

Where:
ε is the induced emf,
N is the number of turns in the loop,
ΔΦ is the change in magnetic flux, and
Δt is the change in time.

In this case, the loop is moving through a uniform magnetic field, so the magnetic flux changes as the loop moves. The magnetic flux is given by the equation:

Φ = B * A

Where:
Φ is the magnetic flux,
B is the magnetic field, and
A is the area of the loop.

Since the magnetic field outside this region is zero, the change in magnetic flux is equal to the initial magnetic flux of the loop multiplied by the magnetic field:

ΔΦ = B * A

To calculate the area of the loop, we need to know the dimensions of the loop. If the length of the loop is L and the width is W, then:

A = L * W

Given the speed of the loop, we can calculate the time it takes to leave the region:

Δt = L / v

Now we can substitute these values into the equation for Faraday's Law:

ε = -N * ΔΦ/Δt
ε = -N * (B * A) / (L / v)

Now we have all the necessary information to calculate the magnitude of the change in the magnetic flux.