Six charges +q, -q, +q, -q, +q and -q are fixed at the corners of a hexagon of side d as shown in the figure.

Question

Six charges +q, -q, +q, -q, +q and -q are fixed at the corners of a hexagon of side d as shown in the figure. The work done in bringing a charge q₀ to the centre of the hexagon from infinity is: (ε₀ - permittivity of free space)

Accepted Answer

1. **Electric Potential:** The electric potential ($V$) is a scalar quantity. The total potential at the center of the hexagon is the algebraic sum of the potentials due to individual charges.
2. **Geometry:** For a regular hexagon of side $d$, the distance from each corner to the center is also $d$.
3. **Calculation:**
   $V_{center} = \frac{1}{4\pi\varepsilon_0} \sum \frac{Q_i}{r_i} = \frac{1}{4\pi\varepsilon_0 d} (q - q + q - q + q - q)$
   $V_{center} = \frac{1}{4\pi\varepsilon_0 d} (0) = 0$.
4. **Work Done:** The work done ($W$) to bring a charge $q_0$ from infinity (where potential is zero) to the center is:
   $W = q_0(V_{center} - V_{\infty}) = q_0(0 - 0) = 0$.

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