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
To determine the magnitude of the induced electric field at a position 8.2 cm from the solenoid's axis, we can use Faraday's law of electromagnetic induction. This law states that the rate of change of magnetic flux through a closed loop is proportional to the induced electromotive force (emf) in that loop.
The magnetic flux density (B) inside a long solenoid can be approximated by B = μ₀ * n * I, where μ₀ is the permeability of free space, n is the number of turns per meter, and I is the current flowing through the solenoid.
The magnetic flux (Φ) through a circular area perpendicular to the solenoid's axis is given by Φ = B * A, where A is the area of the circle.
The induced emf (ε) can be calculated using Faraday's law as ε = -dΦ/dt, where dt is the change in time.
The induced electric field (E) at a distance r from the solenoid's axis is related to the induced emf by ε = -dΦ/dt = E * 2πr, assuming a circular loop.
To find the magnitude of the induced electric field, we need to find the value of ε and divide it by 2πr.
Given:
- Diameter of the solenoid, d = 12 cm, which gives a radius r = 6 cm = 0.06 m
- Number of turns per meter, n = 1200 turns/m
- Initial current, I = 20 A
- Rate of current change, di/dt = -4.33 A/s
- Distance from the solenoid's axis, r = 8.2 cm = 0.082 m
Now, we are ready to calculate the magnitude of the induced electric field:
1. Calculate the magnetic flux density inside the solenoid:
B = μ₀ * n * I
2. Calculate the change in magnetic flux through the circular loop:
dΦ = B * A
A = π * r²
3. Calculate the rate of change of magnetic flux:
dΦ/dt = dΦ / dt
4. Calculate the induced electromotive force:
ε = - dΦ/dt
5. Finally, calculate the magnitude of the induced electric field:
E = ε / (2πr)
By following these steps and plugging in the given values, you can find the magnitude of the induced electric field at a position 8.2 cm from the solenoid's axis.