Let's break down the problem:

We have a system of four charges, $+q$, $-q$, $+2q$, and $-2q$, placed along a straight line. The distances between them are specified:

• The distance between $+q$ and $-q$ is $2l$.
• The distance between $+2q$ and $-2q$ is $4l$.

We need to find the electrostatic potential at point $P$, which is at an angle of $60^\circ$ to the horizontal axis, and is located a distance $r$ from the center $O$.

The potential at a point due to a charge $q$ is given by:

$V = \frac{1}{4 \pi \varepsilon_0} \frac{q}{r}$

This constant is given as:

$\frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \, \text{N m}^2/\text{C}^2$

We are tasked with determining the value of $\alpha$ in the expression for the potential at point $P$:

$V_P = -\alpha \left(\frac{q}{r}\right) \times 10^9 \, \text{V}$

Key Steps to Solve the Problem:

1. Positioning the Charges:

   • Charges $+q$, $-q$, $+2q$, and $-2q$ are placed symmetrically along the horizontal axis.
   • The distances between the charges and point $P$ need to be calculated using geometry and trigonometry.

2. Electrostatic Potential at Point $P$:

   • Calculate the potential due to each charge at point $P$ using the formula for electrostatic potential and sum up their contributions.
   • Consider the angles and distances involved to properly account for the contributions of each charge.

3. Symmetry and Charge Interaction:

   • The symmetrical arrangement will cause some of the potentials to cancel out partially, simplifying the sum of potentials at point $P$.

Let's work through the math step-by-step.

I will now calculate it.The total potential at point $P$ sums to zero. This result occurs due to the symmetry of the charge arrangement. The contributions from the positive and negative charges cancel each other out, resulting in no net potential at point $P$.

Key Points:

• The charges $+q$ and $-q$ cancel each other’s contributions.
• Similarly, the charges $+2q$ and $-2q$ also cancel each other’s effects.
• This is a typical result in symmetric charge distributions.

Now, the value of $\alpha$ in the expression for the potential must also be zero, since the overall potential at $P$ is zero.

Do you want more details or have any other questions?

Related Questions:

1. How would the potential change if one of the charges was moved slightly from the symmetric position?
2. What happens to the electric field at point $P$ due to these charges?
3. How does the potential change if point $P$ is shifted closer to one of the charges?
4. Can we calculate the electric potential at other points on the line joining the charges?
5. What would be the effect on potential if all charges were doubled?

Tip:

In problems with symmetrical charge arrangements, always check for potential cancellations before calculating.