To determine the deflection of the compass needle placed at the center of a circular coil, we can use the principles of electromagnetism. The coil generates a magnetic field when an electric current flows through it, and this magnetic field interacts with the Earth's magnetic field, causing the compass needle to deflect. Let's break down the calculation step by step.
Understanding the Magnetic Field of the Coil
The magnetic field (B) at the center of a circular coil can be calculated using the formula:
B = (μ₀ * n * I) / 2
Where:
- B = magnetic field at the center of the coil (in teslas)
- μ₀ = permeability of free space (4π × 10-7 T·m/A)
- n = number of turns per unit length (in turns/m)
- I = current flowing through the coil (in amperes)
Calculating the Number of Turns per Unit Length
Since we have a circular coil with 49 turns, we can express the number of turns per unit length as:
n = Total turns / Length of the coil
The length of the coil can be calculated using the circumference formula:
Length = 2πr
Given that the radius (r) is 8 cm (which is 0.08 m), we can calculate:
Length = 2π(0.08) ≈ 0.5027 m
Now we can find n:
n = 49 turns / 0.5027 m ≈ 97.4 turns/m
Calculating the Magnetic Field at the Center of the Coil
Now we can substitute the values into the magnetic field formula:
B = (4π × 10-7 T·m/A * 97.4 turns/m * 0.8 A) / 2
Calculating this gives:
B ≈ (4π × 10-7 * 97.4 * 0.8) / 2 ≈ 4.9 × 10-5 T
Finding the Deflection of the Compass Needle
The deflection (θ) of the compass needle can be found using the tangent of the angle, which relates the magnetic field of the coil to the Earth's magnetic field:
tan(θ) = B / Bearth
Where:
- B = magnetic field due to the coil (4.9 × 10-5 T)
- Bearth = horizontal intensity of the Earth's magnetic field (22.48 × 10-6 T)
Substituting the values:
tan(θ) = (4.9 × 10-5) / (22.48 × 10-6)
Calculating this gives:
tan(θ) ≈ 2.18
Calculating the Angle of Deflection
To find the angle θ, we take the arctangent:
θ = arctan(2.18)
Using a calculator, we find:
θ ≈ 65.4°
Summary of Results
The compass needle will deflect approximately 65.4 degrees due to the magnetic field generated by the current flowing through the coil. This interaction between the coil's magnetic field and the Earth's magnetic field illustrates the principles of electromagnetism and how electric currents can create magnetic effects.
