To determine the magnetic flux linked with the larger loop due to the current flowing through the smaller loop, we can use the principles of magnetism, specifically the Biot-Savart Law and the concept of magnetic flux. Let's break this down step by step.
Understanding Magnetic Field from a Current Loop
When current flows through a circular loop, it generates a magnetic field around it. The magnetic field (B) at a point along the axis of a circular loop can be calculated using the formula:
B = (μ₀/4π) * (2πR² * I) / (R² + z²)^(3/2)
Here:
- μ₀ is the permeability of free space, approximately 4π x 10-7 T·m/A.
- R is the radius of the smaller loop (0.003 m).
- I is the current flowing through the smaller loop (2.0 A).
- z is the distance from the center of the smaller loop to the point where we want to calculate the magnetic field (15 cm = 0.15 m).
Calculating the Magnetic Field at the Larger Loop
Substituting the values into the formula:
B = (μ₀/4π) * (2π(0.003)² * 2.0) / ((0.003)² + (0.15)²)^(3/2)
First, calculate the denominator:
(0.003)² + (0.15)² = 0.000009 + 0.0225 = 0.022509
Now raise this to the power of 3/2:
(0.022509)^(3/2) = 0.0001068 (approximately)
Now plug in the values:
B = (4π x 10-7) / (4π) * (0.000018 * 2.0) / 0.0001068
This simplifies to:
B ≈ (10-7) * (0.000036) / 0.0001068
Calculating this gives:
B ≈ 3.37 x 10-5 T
Calculating the Magnetic Flux through the Larger Loop
The magnetic flux (Φ) linked with a loop is given by the formula:
Φ = B * A
Where A is the area of the larger loop. The area of a circle is calculated using:
A = πR²
For the larger loop with a radius of 20 cm (0.2 m):
A = π(0.2)² = π(0.04) ≈ 0.1256 m²
Now, substituting the values into the flux equation:
Φ = (3.37 x 10-5 T) * (0.1256 m²)
Calculating this gives:
Φ ≈ 4.23 x 10-6 Wb
Final Result
Thus, the magnetic flux linked with the larger loop due to the current in the smaller loop is approximately 4.23 x 10-6 Weber. This value illustrates how even a small current in a small loop can influence a much larger loop, showing the interconnectedness of magnetic fields in different sizes of circuits.
