• how is electric potential at a point due to a charge q equal to q/4.pi.epsilon not.dist between the charges

To understand how the electric potential at a point due to a charge \( q \) is expressed as \( V = \frac{q}{4 \pi \epsilon_0 r} \), we need to break down the concepts of electric potential, charge, and the role of distance in this relationship.

Defining Electric Potential

Electric potential, often denoted as \( V \), is a measure of the potential energy per unit charge at a specific point in an electric field. It tells us how much work would be done to move a unit positive charge from a reference point (usually infinity) to that point in the field without any acceleration.

The Role of Charge

The charge \( q \) creates an electric field around it. The strength of this field diminishes with distance, which is a fundamental property of point charges. The electric potential at a distance \( r \) from a point charge is directly proportional to the amount of charge and inversely proportional to the distance from the charge.

Understanding the Formula

The formula for electric potential due to a point charge is given by:

V = \frac{q}{4 \pi \epsilon_0 r}

Here’s what each term represents:

• q: The magnitude of the point charge creating the electric potential.
• r: The distance from the charge to the point where the potential is being measured.
• \(\epsilon_0\): The permittivity of free space, a constant that describes how electric fields interact with the vacuum of space.
• 4π: This factor arises from the geometry of three-dimensional space, specifically the surface area of a sphere.

Why the Inverse Relationship with Distance?

The inverse relationship with distance \( r \) indicates that as you move further away from the charge, the electric potential decreases. This makes intuitive sense: the further you are from a charge, the less influence it has on you. Imagine standing near a loudspeaker; the sound is much louder when you are close compared to when you are far away.

Deriving the Formula

The derivation of this formula involves integrating the electric field created by the charge. The electric field \( E \) due to a point charge is given by:

E = \frac{q}{4 \pi \epsilon_0 r^2}

To find the electric potential \( V \), we consider the work done in moving a charge \( q_0 \) from infinity to a point at distance \( r \). The work done \( W \) is the integral of the electric field over distance:

W = -\int_{\infty}^{r} E \, dr = -\int_{\infty}^{r} \frac{q}{4 \pi \epsilon_0 r^2} \, dr

Evaluating this integral gives us the expression for electric potential:

V = \frac{q}{4 \pi \epsilon_0 r}

Practical Implications

This formula is crucial in various applications, from designing electrical circuits to understanding phenomena in electrostatics. For instance, if you know the charge and the distance, you can easily calculate the potential at any point in the field, which is essential for predicting how charges will behave in that field.

In summary, the electric potential due to a point charge is a fundamental concept in electrostatics, encapsulated in the formula \( V = \frac{q}{4 \pi \epsilon_0 r} \). It highlights the relationship between charge, distance, and the resulting potential energy, providing a foundation for understanding electric fields and forces.