Magnetic Field at the Center of a Circular Coil

We have a circular coil of wire with:

• Number of turns: $N = 100$
• Radius of each turn: $R = 8.0\ \text{cm} = 0.080\ \text{m}$
• Current in the coil: $I = 0.40\ \text{A}$

We want the magnetic field magnitude $B$ at the center of the coil.

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1. Formula for a Single Circular Loop

For a single circular loop of radius $R$ carrying current $I$, the magnetic field at the center is:

$$B_1 = \frac{\mu_0 I}{2R}$$

where $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$ is the permeability of free space.

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2. Coil with $N$ Turns

If there are $N$ closely spaced turns, each carrying the same current $I$ and having the same radius $R$, the total field at the center is the sum of the contributions from all turns:

$$B = N \cdot B_1 = N \cdot \frac{\mu_0 I}{2R}$$

Substitute the known values:

• $N = 100$
• $I = 0.40\ \text{A}$
• $R = 0.080\ \text{m}$
• $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$

So:

$$B = 100 \cdot \frac{(4\pi \times 10^{-7}) (0.40)}{2 \times 0.080}$$

Simplify step by step:

1. Denominator: $2R = 2 \times 0.080 = 0.16\ \text{m}$
2. Numerator inside: $(4\pi \times 10^{-7}) \times 0.40 = 1.6\pi \times 10^{-7}$
3. Fraction for one turn: $B_1 = \frac{1.6\pi \times 10^{-7}}{0.16} = 10\pi \times 10^{-7}\ \text{T} = \pi \times 10^{-6}\ \text{T}$
4. Multiply by $N = 100$: $B = 100 \cdot (\pi \times 10^{-6}) = 100\pi \times 10^{-6}\ \text{T} = \pi \times 10^{-4}\ \text{T}$

Using $\pi \approx 3.14$:

$$B \approx 3.14 \times 10^{-4}\ \text{T}$$

So, the magnitude of the magnetic field at the center of the coil is:

$$\boxed{B \approx 3.1 \times 10^{-4}\ \text{T}}$$

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3. What the Visualization Shows

The canvas visualization treats the coil as a glowing circular loop (with many turns implied by line thickness and color), centered on the screen. You can interactively change:

• The radius of the coil (in centimeters)
• The current through the coil (in amperes)
• The number of turns

The computed magnetic field at the center is displayed as text and also encoded visually:

• A vertical magnetic-field vector at the center, whose length scales with $|B|$.
• A halo or glow around the center, whose intensity and size grow with $|B|$.

Mathematically, the field used in the visualization is:

$$B = N\,\frac{\mu_0 I}{2R}, \quad R = (\text{radius in cm})/100$$

You can see how increasing $N$ or $I$ boosts the field, while increasing $R$ (making the coil larger) weakens it. The animation also gently rotates phase coloring around the loop to suggest the circulation of current, while the central arrow stays fixed to represent the static direction of $\vec{B}$ (out of or into the screen, here represented in 2D as an upward arrow).