Show that the magnetic field produced at the centre of a regular hexagon carrying a steady current 𝐼 = 6 × 102 [mA] as a function of 𝑅, the distance from the centre of the polygon to any side, is
|H ̅ |=3I/(π R)  sin⁡(π/6)
Be sure to show a picture of the problem. Plot the value of the fields as a function of 𝑅 being sure to label/explain the units and any key features of the plot.

The plot above shows the magnetic field \( |\mathbf{H}| \) at the center of the hexagon as a function of \( R \), the distance from the center to any side. The key features of the plot are:

1. The magnetic field \( |\mathbf{H}| \) decreases as the distance \( R \) increases.
2. The field falls off as \( 1/R \), which is expected from the relationship \( |\mathbf{H}| \propto 1/R \).
3. For small values of \( R \), the magnetic field is stronger, while it diminishes significantly as \( R \) grows larger.
4. The units of the magnetic field \( |\mathbf{H}| \) are in Amperes per meter (A/m), while \( R \) is in meters.

Would you like more details on the derivation, or do you have any further questions?

Here are some additional related questions for deeper understanding:
1. How does the magnetic field change if the current \( I \) is doubled?
2. What would happen to the magnetic field if the hexagon had more sides (e.g., an octagon)?
3. How does the direction of the magnetic field vary for different points inside the hexagon?
4. How does the formula for the magnetic field change for an irregular polygon?
5. What is the significance of using the sine function \( \sin(\pi/6) \) in the magnetic field formula?

**Tip:** In problems involving current-carrying loops or polygons, symmetry can greatly simplify the calculation of the magnetic field.

Show that the magnetic field produced at the centre of a regular hexagon carrying a steady current I = 6 × 10^2 [mA] as a function of R, the distance from the centre of the polygon to any side, is |H| = 3I/(π R) sin(π/6). Be sure to show a picture of the problem. Plot the value of the fields as a function of R, being sure to label/explain the units and any key features of the plot.

Learn how to calculate the magnetic field at the center of a regular hexagon carrying a steady current. Using the formula |H| = 3I/(πR) * sin(π/6), this problem involves principles from electromagnetism, symmetry, and trigonometry. Includes a plot showing the magnetic field as a function of the distance R, with key features explained. Suitable for undergraduate physics students.

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