A 543-Turn Circular-Loop Coil and Earth's Magnetic Field

Calculating Earth's Magnetic Field with a Circular-Loop Coil

A 543-turn circular-loop coil with a diameter of 16.5 cm is initially aligned so that its plane is perpendicular to the Earth's magnetic field. In 2.42 ms the coil is rotated 90.0∘ so that its plane is parallel to the Earth's magnetic field. An average emf of 0.149 V is induced in the coil. What is the value of the Earth's magnetic field?

Question:

What is the value of the Earth's magnetic field when a 543-turn circular-loop coil is rotated from perpendicular to parallel to the Earth's magnetic field, inducing an average emf of 0.149 V in the coil?

Answer:

The value of the Earth's magnetic field will be approximately 0.220 Tesla.

Faraday's law of electromagnetic induction states that the electromotive force (emf) induced in a loop is proportional to the rate of change of magnetic flux through the loop. The formula for calculating the induced emf is:

emf = -N × ΔΦ / Δt

Where:

emf is the induced electromotive force

N is number of turns in the coil

ΔΦ is change in magnetic flux through the coil

Δt is the time over which the change occurs

In this case, the coil is initially perpendicular to the Earth's magnetic field, and then it's rotated 90° to be parallel to the magnetic field. The change in magnetic flux will be calculated as:

ΔΦ = B × A × cos(θ_initial) - B × A × cos(θ_final)

Where:

B is the magnetic field strength

A is the area of the coil

θ_initial is the initial angle between the coil's plane and the magnetic field

θ_final is the final angle between the coil's plane and the magnetic field

Given:

emf = 0.149 V

N = 543

Diameter (D) = 16.5 cm = 0.165 m (converted to meters)

Time (Δt) = 2.42 ms = 2.42 × 10⁻³ s (converted to seconds)

θ_initial = 90° (perpendicular to the magnetic field)

θ_final = 0° (parallel to the magnetic field)

Substitute the values and rearrange the formula to solve for B:

B = -emf / (N × A × Δt × (cos(θ_initial) - cos(θ_final)))

The area of the circular coil can be calculated using the formula A = π × (D/2)².

Substitute the values and calculate:

A = π × (0.165 m / 2)² ≈ 0.021237 m²

cos(90°) = 0

cos(0°) = 1

B = -0.149 V / (543 × 0.021237 m² × 2.42 × 10⁻³ s × (0 - 1))

B ≈ 0.220 T

Therefore, the value of the Earth's magnetic field is approximately 0.220 Tesla.