A length of wire carrying a current i and bent into a circular coil of one turn.The same length of wire has been out to give a coil of 5 turns, each of 1\5 the original radius.If Ba and Bb are the magnitudes of the magnetic fields at the centers of the two coils, what is the ratio Bb\Ba?

The formula you need can be found here:

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/curloo.html#c2

Multiply it by N if there are N loops.

Since the field at the center is inversely proportional to R, The field increases by N^2 . The ratio is 25.

To find the ratio Bb/Ba, we need to calculate the magnetic fields at the centers of the two coils.

The magnetic field at the center of a circular loop of wire carrying a current can be calculated using Ampere's law. According to Ampere's law, the magnetic field at the center of a loop is given by:

B = (μ₀ * i) / (2 * r)

Where:
B is the magnetic field at the center of the loop,
μ₀ is the permeability of free space,
i is the current passing through the wire, and
r is the radius of the loop.

Let's calculate the magnetic fields at the centers of the two coils and find their ratio.

For the original coil with one turn:
B₁ = (μ₀ * i) / (2 * r₁)

For the new coil with 5 turns and 1/5 the radius of the original one:
B₂ = (μ₀ * i) / (2 * r₂)

To find the ratio Bb/Ba, we need to calculate Bb and Ba. Bb represents the magnetic field at the center of the coil with 5 turns, and Ba represents the magnetic field at the center of the coil with one turn.

Substituting the given information:
r₂ = r₁ / 5 (1/5 the radius of the original coil)

Calculating Bb:
Bb = (μ₀ * i) / (2 * r₂) = (μ₀ * i) / (2 * (r₁ / 5)) = (5 * μ₀ * i) / (2 * r₁)

Calculating Ba:
Ba = (μ₀ * i) / (2 * r₁)

Now, we can calculate the ratio Bb/Ba:
Bb/Ba = (5 * μ₀ * i) / (2 * r₁) / ((μ₀ * i) / (2 * r₁))
= (5 * μ₀ * i) / (2 * r₁) * (2 * r₁) / (μ₀ * i)
= 5

Therefore, the ratio Bb/Ba is 5.

To find the ratio Bb/Ba, we need to determine the magnetic fields at the centers of the two coils.

1. Magnetic field at the center of the original coil (Ba):
The magnetic field at the center of the original coil can be determined using Ampere's Law. The equation for the magnetic field at the center of a circular coil is given by:
Ba = μ₀ * (N * i) / (2 * R),
where μ₀ is the permeability of free space, N is the number of turns, i is the current, and R is the radius of the coil.

2. Magnetic field at the center of the new coil (Bb):
In the new coil, the radius is reduced to 1/5 of the original radius. Therefore, the radius of the new coil is R/5.
The number of turns remains the same (5 turns).
Using the same formula as above, the magnetic field at the center of the new coil can be calculated as:
Bb = μ₀ * (N * i) / (2 * (R/5)).

Now, let's substitute the values into the formulas:

Ba = μ₀ * (1 * i) / (2 * R)
Bb = μ₀ * (5 * i) / (2 * (R/5))

To find the ratio Bb/Ba, we divide Bb by Ba:

Bb/Ba = (μ₀ * (5 * i) / (2 * (R/5))) / (μ₀ * (1 * i) / (2 * R))
= (5 * i * 2 * R) / (2 * R * 1 * i)
= 5

Therefore, the ratio Bb/Ba is 5.