Two identical coils carrying equal current have a common centre, and their planes are at right angles to each other. Find the ratio of the magnitude of the resultant magnetic field at centre and the field due to one coil alone

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Answer

Solution:

Given:

Two identical coils carrying equal current have a common centre, and their planes are at right angles to each other.

To find:

The ratio of the magnitude of the resultant magnetic field at centre and the field due to one coil alone.

Explanation:

When two identical coils carrying equal current have a common centre, and their planes are at right angles to each other, the magnetic field at the centre of the coils can be found as follows:

1. Magnetic field due to one coil:
The magnetic field at the centre of one coil can be found using the formula:

B = (μ₀/4π) × (2I/R)

Where,
B = Magnetic field at the centre of the coil
μ₀ = Permeability of free space = 4π × 10⁻⁷ T m/A
I = Current flowing through the coil
R = Radius of the coil

2. Magnetic field due to one coil:
The magnetic field at the centre of the two coils can be found using the formula:

B' = (μ₀/4π) × (4I/R)

Where,
B' = Magnetic field at the centre of the two coils
μ₀ = Permeability of free space = 4π × 10⁻⁷ T m/A
I = Current flowing through each coil
R = Radius of each coil

3. Resultant magnetic field:
The resultant magnetic field at the centre of the two coils can be found using the Pythagorean theorem:

B_res = √(B² + B'²)

4. Ratio of the magnitude of the resultant magnetic field and the field due to one coil alone:
The ratio of the magnitude of the resultant magnetic field at centre and the field due to one coil alone can be found using the formula:

Ratio = B_res / B

Answer:

The ratio of the magnitude of the resultant magnetic field at centre and the field due to one coil alone is √2 : 1.

Answer

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