Six charges are placed around a regular hexagon of side length a as shown in the figure Five of them have charge $q$, and the remaining one has charge $x$ The perpendicular from each charge to the nearest hexagon side passes through the center $O$ of the hexagon and is bisected by the side Which of the following statement(s) is(are) correct in SI units?
- When $x=q$, the magnitude of the electric field at $O$ is zero.
- When $x=-q$, the magnitude of the electric field at $O$ is $\frac{q}{6 \pi \in_0 a^2}$.
- When $x=2 q$, the potential at $O$ is $\frac{7 q}{4 \sqrt{3} \pi \in_0 a}$.
- When $x=-3 q$, the potential at $O$ is $\frac{3 q}{4 \sqrt{3} \pi \epsilon_0 a}$.
The Correct Option is C
Approach Solution - 1
When $x=2 q$, the potential at $O$ is $\frac{7 q}{4 \sqrt{3} \pi \in_0 a}$.
Approach Solution -2
Let's review the calculation: Given: - The electric field (\(E\)) at each diagonal is \(E = k \frac{q}{r^2}\). - We have two diagonals forming a \(60^\circ\) angle. - We're looking for the resultant electric field (\(E_{\text{net}}\)). Step-by-Step Solution: 1. Electric Field at a Diagonal (\(E\)): The electric field at each diagonal is given by: \[ E = k \frac{q}{r^2} \] 2. Resultant Electric Field due to Two Diagonals: Considering two diagonals at an angle of \(60^\circ\): - Each diagonal contributes to the electric field with magnitude \(E\). - To find the net electric field, we use the formula for the resultant of two vectors at an angle: \[ E_{\text{net}} = \sqrt{(2E \cos(30^\circ))^2 + (2E \sin(30^\circ))^2} \] \[ E_{\text{net}} = \sqrt{(2kq/\sqrt{3})^2 + (2kq)^2} \] \[ E_{\text{net}} = \sqrt{(4k^2q^2/3) + (4k^2q^2)} \] \[ E_{\text{net}} = \sqrt{(4k^2q^2/3 + 12k^2q^2/3)} \] \[ E_{\text{net}} = \sqrt{(16k^2q^2/3)} \] \[ E_{\text{net}} = \frac{4kq}{\sqrt{3}} \] 3. Simplify the Expression: Using the relationship \(k = \frac{1}{4\pi\epsilon_0}\): \[ E_{\text{net}} = \frac{q}{\pi\epsilon_0} \] Conclusion: The resultant electric field (\(E_{\text{net}}\)) due to two diagonals at an angle of \(60^\circ\) in the symmetrical hexagon is \( \frac{q}{\pi\epsilon_0}\).
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Concepts Used:
Electrostatic Potential and Capacitance
Electrostatic Potential
The potential of a point is defined as the work done per unit charge that results in bringing a charge from infinity to a certain point.
Some major things that we should know about electric potential:
- They are denoted by V and are a scalar quantity.
- It is measured in volts.
Capacitance
The ability of a capacitor of holding the energy in form of an electric charge is defined as capacitance. Similarly, we can also say that capacitance is the storing ability of capacitors, and the unit in which they are measured is “farads”.
Read More: Electrostatic Potential and Capacitance
The capacitor is in Series and in Parallel as defined below;
In Series
Both the Capacitors C1 and C2 can easily get connected in series. When the capacitors are connected in series then the total capacitance that is Ctotal is less than any one of the capacitor’s capacitance.
In Parallel
Both Capacitor C1 and C2 are connected in parallel. When the capacitors are connected parallelly then the total capacitance that is Ctotal is any one of the capacitor’s capacitance.