Answer:
a) Фm = μ₀*I*n*π*N*R₁²
b) Фm = μ₀*I*n*π*N*R₂²
Explanation:
Given
n = number of turns per unit length of the solenoid
R₁ = radius of the solenoid
I = current passing through the solenoid
R₂ = radius of the circular coil
N = number of total turns on the coil
a) Фm = ? If R₂ > R₁
b) Фm = ? If R₂ < R₁.
We can use the formula
Фm = N*B*S*Cos θ
where
N is the number of turns
B is the magnetic field
S is the area perpendicular to magnetic field B.
The magnetic field outside the solenoid can be approximated to zero. Therefore, the flux through the coil is the flux in the core of the solenoid.
We know that the magnetic field inside the solenoid is uniform. Thus, the flux through the circular coil is given by the same expression with R2 replacing R1 (from the area).
a) Then, the flux through the large circular loop outside the solenoid (R₂ > R₁) is obtained as follows:
Фm = N*B*S*Cos θ
where
B = μ₀*I*n
S = π*R₁²
θ = 0°
⇒ Фm = (N)*(μ₀*I*n)*(π*R₁²)*Cos 0°
⇒ Фm = μ₀*I*n*π*N*R₁²
b) The flux through the coil when R₂ < R₁ is
Фm = (N)*(μ₀*I*n)*(π*R₂²)*Cos 0°
⇒ Фm = μ₀*I*n*π*N*R₂²
The answer is D.
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Hope this helps!