A length of copper wire is shaped into a single circular loop of circumference 63 cm, similar to that shown in the picture below.
A magnetic field of strength 0.50 T is parallel to the normal of the loop. The copper wire has a resistance of 2.5 x 10-2 Ω.
The ends of the wire loop are pulled suddenly so that the new circumference is reduced to 0.7 times its original circumference, in 400 ms.
Determine the magnitude of the induced current that flows in the copper wire.
To determine the magnitude of the induced current that flows in the copper wire, we can use Faraday's law of electromagnetic induction. According to this law, the induced electromotive force (emf) in a loop of wire is equal to the rate of change of magnetic flux through the loop.
1. First, let's calculate the initial magnetic flux through the loop of wire. The magnetic flux is given by the product of the magnetic field strength (B) and the area (A) enclosed by the loop. The magnetic field is parallel to the normal of the loop, so the area is equal to the enclosed area of the circle.
The initial circumference of the loop is 63 cm, so the initial radius (r) can be calculated by dividing the circumference by 2π:
r = 63 cm / (2π).
Convert the radius to meters: r = (63 cm / (2π)) / 100.
The initial area of the loop is given by the formula: A = πr^2.
Now, calculate the initial magnetic flux (Φ_initial): Φ_initial = B * A.
2. Next, let's calculate the final magnetic flux through the loop of wire. The final circumference is 0.7 times the original circumference, so the final radius (r_final) can be calculated similarly: r_final = (0.7 * 63 cm / (2π)) / 100.
Calculate the final area (A_final) of the loop using the final radius: A_final = π * r_final^2.
Finally, calculate the final magnetic flux (Φ_final): Φ_final = B * A_final.
3. The rate of change of magnetic flux (∆Φ/∆t) is the difference between the final and initial magnetic flux divided by the time taken (∆t). In this case, the time taken is given as 400 ms, which should be converted to seconds: ∆t = 400 ms / 1000.
Calculate the rate of change of magnetic flux: ∆Φ/∆t = (Φ_final - Φ_initial) / ∆t.
4. According to Faraday's law, the induced emf (∆V) is equal to the rate of change of magnetic flux (∆Φ/∆t) multiplied by the number of turns in the wire loop (N). In this case, there is only one turn, so N = 1.
Calculate the induced emf (∆V) = ∆Φ/∆t.
5. Finally, we can use Ohm's law to calculate the magnitude of the induced current (I). The induced emf (∆V) is equal to the product of the induced current (I) and the resistance (R) of the wire: ∆V = I * R.
Rearrange the equation to solve for the magnitude of the induced current (I) = ∆V / R.
Substitute the calculated values into the equations and calculate the magnitude of the induced current that flows in the copper wire.
To determine the magnitude of the induced current that flows in the copper wire, we can use Faraday's law of electromagnetic induction.
Step 1: Calculate the change in magnetic flux
The change in magnetic flux is given by the equation:
ΔΦ = B * ΔA * cos(θ)
Here, B is the magnetic field strength, ΔA is the change in area enclosed by the loop, and θ is the angle between the magnetic field and the normal to the loop. In this case, θ = 0 degrees since the magnetic field is parallel to the normal of the loop.
Given that B = 0.50 T, we need to determine the change in area, ΔA.
Step 2: Calculate the change in area
The original circumference of the loop is given as 63 cm. The new circumference is reduced to 0.7 times its original circumference. So, the new circumference is 0.7 * 63 cm.
The radius of the original loop is given by:
r = C / (2π)
where C is the circumference.
So, the radius of the original loop is:
r = 63 cm / (2π)
The radius of the new loop is:
r' = (0.7 * 63 cm) / (2π)
The change in area is given by:
ΔA = π * (r'^2 - r^2)
Step 3: Calculate the change in magnetic flux
Using the value of B = 0.50 T and ΔA obtained in Step 2, we can calculate the change in magnetic flux, ΔΦ.
Step 4: Calculate the induced emf
According to Faraday's law of electromagnetic induction, the induced emf (ε) is equal to the negative rate of change of magnetic flux.
ε = -ΔΦ / Δt
Given that Δt = 400 ms, we can calculate the induced emf.
Step 5: Calculate the induced current
The induced current (I) can be calculated using Ohm's law, where the resistance (R) is given as 2.5 x 10^-2 Ω.
I = ε / R
Given all the values necessary, we can calculate the magnitude of the induced current.