A long, ideal solenoid has a diameter d=12cm and n=1200turns/meter carrying current I=20A. If the current is lowered at 4.33amp/s to zero, what is the magnitude of the induced electric field in V/m at a position 8.2cm from the solenoid's axis (so at a point outside the solenoid)?
To find the magnitude of the induced electric field at a point outside the solenoid, we can use Faraday's law of electromagnetic induction. This law states that the induced electromotive force (EMF) in a closed loop is equal to the negative rate of change of the magnetic flux through that loop.
The magnetic field inside a long, ideal solenoid is given by the equation B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length (turns/meter), and I is the current in the solenoid.
The magnetic flux through a circular loop at a distance r from the solenoid's axis can be calculated using the equation Φ = B*A, where Φ is the magnetic flux, B is the magnetic field, and A is the area of the loop.
In this case, the solenoid has a diameter of 12 cm, so its radius is 6 cm (or 0.06 m). The area of the loop can be calculated as A = π*r².
Now, let's find the initial magnetic flux (Φ₀) when the current is 20 A. We'll use the given values of d = 12 cm, n = 1200 turns/meter, I = 20 A, and r = 8.2 cm (or 0.082 m).
Radius (r) = 0.082 m
Area (A) = π*(0.082)^2 m²
Now, we can calculate the initial magnetic flux:
Φ₀ = B*A₀
= (μ₀nI)*(π*(0.082)^2) T·m²
Next, we need to calculate the final magnetic flux (Φ₁) when the current decreases to zero. Since the current is changing at a rate of 4.33 A/s, the time required to reach zero current can be calculated using the equation Δt = I/f, where Δt is the time, I is the initial current, and f is the rate of change of current.
Let's calculate the time (Δt):
Δt = I/f
= 20 A / 4.33 A/s
Now, using Faraday's law, the induced electromotive force (EMF) is given by the equation EMF = -(dΦ/dt), where EMF is the induced electromotive force, Φ is the magnetic flux, and t is time.
Finally, to find the magnitude of the induced electric field (E) at the given position outside the solenoid, we can use the equation E = EMF/r, where E is the induced electric field and r is the distance from the solenoid's axis.
By following these calculations, you can find the magnitude of the induced electric field at a position 8.2 cm from the solenoid's axis when the current is lowered at 4.33 A/s to zero.