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 2.2cm from the solenoid's axis (so at a point inside the solenoid)?
To determine the magnitude of the induced electric field (E) at a point inside the solenoid, you can use Faraday's law of electromagnetic induction. According to this law, the magnitude of the induced electric field is given by the equation:
E = -N * dϕ/dt
Where:
- E is the induced electric field.
- N is the number of turns per unit length of the solenoid.
- dϕ/dt is the rate of change of the magnetic flux through a surface with respect to time.
Now, let's break down the steps to find the magnitude of the induced electric field:
Step 1: Calculate the magnetic flux (ϕ) through a surface inside the solenoid.
The magnetic field inside an ideal solenoid is given by the equation:
B = μ₀ * N * I
Where:
- B is the magnetic field.
- μ₀ is the permeability of free space.
- N is the number of turns per unit length of the solenoid.
- I is the current passing through the solenoid.
The magnetic flux (ϕ) passing through a surface inside the solenoid is given by the equation:
ϕ = B * A
Where:
- ϕ is the magnetic flux.
- B is the magnetic field.
- A is the area of the surface.
Substituting the values, we have:
ϕ = (μ₀ * N * I) * A
Step 2: Calculate the rate of change of the magnetic flux (dϕ/dt).
Given that the current is decreasing at a rate of 4.33 A/s, the rate of change of the magnetic flux is:
dϕ/dt = (μ₀ * N * dI/dt) * A
Step 3: Calculate the induced electric field (E).
Substituting the values into the formula:
E = -N * dϕ/dt
= -N * (μ₀ * N * dI/dt) * A
Step 4: Plug in the values provided in the question.
- Diameter (d) = 12 cm = 0.12 m (since 1 cm = 0.01 m)
- Number of turns per meter (N) = 1200 turns/m
- Current (I) = 20 A
- Rate of change of current (dI/dt) = -4.33 A/s (negative sign indicates decreasing current)
- Distance from the solenoid's axis (r) = 2.2 cm = 0.022 m
We need to calculate A, the area of the surface inside the solenoid.
Step 5: Calculate the area (A).
The area of the surface inside the solenoid can be calculated using the formula:
A = π * r^2
Substituting the value of r, we have:
A = π * (0.022)^2
Step 6: Calculate the induced electric field (E) using the given values.
Substituting all the values into the formula for E:
E = -N * (μ₀ * N * dI/dt) * A
= -1200 * (4π * 10^-7 T·m/A) * 1200 * (-4.33 A/s) * π * (0.022)^2
Finally, calculate the magnitude (absolute value) of E to find the answer.