Question
What is the electric potential at the intersection of the diagonals of a regular hexagon with equal charges at each corner? The hexagon has side length x.
Answers
GPT-4o mini
To find the electric potential at the intersection of the diagonals of a regular hexagon with equal charges placed at each corner, we will follow these steps:
1. **Identify the position of the center of the hexagon**: The intersection of the diagonals of a regular hexagon coincides with its center.
2. **Electric potential due to a point charge**: The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by:
\[
V = \frac{kQ}{r}
\]
where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
3. **Position of charges**: Denote the charges at the vertices of the hexagon as \( Q_1, Q_2, Q_3, Q_4, Q_5, Q_6 \). Since the charges are equal, we can denote each charge by \( Q \).
4. **Distance from the center to a vertex**: In a regular hexagon with a side length \( x \), the distance from the center to a vertex can be found using trigonometry. Each vertex of the hexagon is located a distance of \( r \) from the center:
\[
r = x
\]
(since the radius of the circumcircle of the hexagon equals the side length for a regular hexagon).
5. **Calculating the total electric potential**: The total electric potential \( V_{\text{total}} \) at the center is the sum of the potentials due to each charge:
\[
V_{\text{total}} = V_1 + V_2 + V_3 + V_4 + V_5 + V_6
\]
Each term contributes equally:
\[
V_{\text{total}} = 6 \cdot \frac{kQ}{r}
\]
where \( r = x \):
\[
V_{\text{total}} = 6 \cdot \frac{kQ}{x}
\]
Therefore, the electric potential at the intersection of the diagonals of the regular hexagon with equal charges at each corner is:
\[
\boxed{\frac{6kQ}{x}}
\]
1. **Identify the position of the center of the hexagon**: The intersection of the diagonals of a regular hexagon coincides with its center.
2. **Electric potential due to a point charge**: The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by:
\[
V = \frac{kQ}{r}
\]
where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
3. **Position of charges**: Denote the charges at the vertices of the hexagon as \( Q_1, Q_2, Q_3, Q_4, Q_5, Q_6 \). Since the charges are equal, we can denote each charge by \( Q \).
4. **Distance from the center to a vertex**: In a regular hexagon with a side length \( x \), the distance from the center to a vertex can be found using trigonometry. Each vertex of the hexagon is located a distance of \( r \) from the center:
\[
r = x
\]
(since the radius of the circumcircle of the hexagon equals the side length for a regular hexagon).
5. **Calculating the total electric potential**: The total electric potential \( V_{\text{total}} \) at the center is the sum of the potentials due to each charge:
\[
V_{\text{total}} = V_1 + V_2 + V_3 + V_4 + V_5 + V_6
\]
Each term contributes equally:
\[
V_{\text{total}} = 6 \cdot \frac{kQ}{r}
\]
where \( r = x \):
\[
V_{\text{total}} = 6 \cdot \frac{kQ}{x}
\]
Therefore, the electric potential at the intersection of the diagonals of the regular hexagon with equal charges at each corner is:
\[
\boxed{\frac{6kQ}{x}}
\]