To determine the potential difference between two points, A and B, in the presence of two point charges, we can use the concept of electric potential. The potential difference, often referred to as voltage, is calculated as the difference in electric potential at these two points, expressed mathematically as:

$$\Delta V = V_B - V_A$$

In this scenario, we have a positive charge \( q \) and a negative charge \( -3q \) positioned along a line. The potential at point B is influenced by both charges, and we can calculate it by summing the contributions from each charge. The formula for electric potential \( V \) due to a point charge is given by:

$$V = k \frac{Q}{r}$$

where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the distance from the charge to the point of interest.

At point B, the potential contributions from the charges are:

1. From charge \( q \): The distance from charge \( q \) to point B is \( s - x \), leading to:

$$V_B(q) = k \frac{q}{s - x}$$

2. From charge \( -3q \): The distance from charge \( -3q \) to point B is \( x \), resulting in:

$$V_B(-3q) = k \frac{-3q}{x}$$

Thus, the total potential at point B is:

$$V_B = k \frac{q}{s - x} - k \frac{3q}{x}$$

Similarly, we calculate the potential at point A. The contributions are:

1. From charge \( q \): The distance from charge \( q \) to point A is \( x \), giving us:

$$V_A(q) = k \frac{q}{x}$$

2. From charge \( -3q \): The distance from charge \( -3q \) to point A is \( s - x \), leading to:

$$V_A(-3q) = k \frac{-3q}{s - x}$$

Thus, the total potential at point A is:

$$V_A = k \frac{q}{x} - k \frac{3q}{s - x}$$

Now, we can find the potential difference by substituting \( V_B \) and \( V_A \) into the equation for \( \Delta V \):

$$\Delta V = \left(k \frac{q}{s - x} - k \frac{3q}{x}\right) - \left(k \frac{q}{x} - k \frac{3q}{s - x}\right)$$

After simplifying, we combine like terms and factor out common factors to arrive at the final expression for the potential difference:

$$\Delta V = \frac{8kqx}{xs - x} - \frac{4kqs}{xs - x}$$

This expression allows us to calculate the potential difference between points A and B based on the positions of the charges and their magnitudes. Understanding these calculations is crucial for solving problems related to electric potential and voltage in electrostatics.