2.2. ELECTROSTATIC POTENTIAL DUE TO A POINT CHARGE

Consider any general static charge configuration. We define potential energy of a test charge q in terms of the work done on the charge q. This work is obviously proportional to q, since the force at any point is qE, where E is the electric field at that point due to the given charge configuration. It is, therefore, convenient to divide the work by the amount of charge q, so that the resulting quantity is independent of q.

Electrostatic potential V is defined based on the work done to move a test charge q in an electric field. Since the force acting on the charge is proportional to the charge itself (F = qE), we can standardize our measurement of work by dividing it by the charge q. This way, the potential we calculate is independent of the specific charge used.

Think of it like measuring the cost of a toll road per car. If the tolls vary with the number of passengers in the car, it becomes complicated. Instead, we can set a flat rate per car regardless of the number of passengers, simplifying the payment process. Electrostatic potential works similarly by normalizing the work done based on charge.

Work done by external force in bringing a unit positive charge from point R to P = V – V = P R.

When we bring a unit positive charge from a point at infinity to a specific point in an electric field, we do work against the electric forces acting on the charge. The work done is expressed as the difference in the electrostatic potential between the two points (P and R). Therefore, the potential difference V represents the work needed per unit positive charge that enables us to calculate the electrostatic potential at any point in an electric field.

Imagine climbing a hill. The work you exert to reach the top (potential energy) is dependent only on the height difference between the starting point and the top. Similarly, the electrostatic potential is about the difference in potential energy per unit charge as you move through the electric field.

If the potential at infinity is chosen to be zero, potential at a point with position vector r due to a point charge Q placed at the origin is given by V(r) = 1/(4πε₀) * (Q/r).

By convention, we define the electric potential at infinity to be zero as a reference point. This allows us to express the potential at any point in space in a more manageable way. When we have a point charge Q, we can calculate the potential V at a distance r from this charge using the formula V(r) = (1/(4πε₀))(Q/r). This relation tells us how strong the electric potential is as we move closer or farther away from the charge.

Consider the sea level as a reference for altitude. Just as we measure how high a mountain is based on its distance above sea level (with sea level being zero), we measure the potential of a charge based on a reference point (infinity) where the potential is zero.

For a point charge Q at the origin, V(r) = 1/(4πε₀) * (Q/r) indicates that as you move away from the charge, the potential decreases with distance.

The formula V(r) = (1/(4πε₀))(Q/r) shows that the electric potential due to a point charge decreases as you move further away from the charge. This inverse relationship indicates that the influence of the charge diminishes with distance, reflecting our intuitive understanding of how gravity works—heavier objects have a larger field of influence which weakens as you step back farther from them.

Think of a light bulb: the brightness of the light diminishes as you move further away from it. A light bulb close to you seems very bright, while at a distance, the brightness lowers. Similarly, the potential from a point charge is strongest at close range and decreases with distance.

Equation (2.8) is true for any sign of the charge Q, though we considered Q > 0 in its derivation. For Q < 0, V < 0, i.e., work done (by the external force) per unit positive test charge in bringing it from infinity to the point is negative.

The potential calculated using V(r) = (1/(4πε₀))(Q/r) applies to both positive and negative charges. For negative charges, the potential becomes negative, indicating that work done by the external force is less than zero, meaning the electric field is acting in the direction of the positive test charge. This highlights an important concept in electric fields: the distinction in the behavior of fields produced by positive versus negative charges.

Consider a gravity hill—the direction of help decreases when moving uphill (adding work), while moving downhill requires less effort. Here, a positive charge moving toward a negative one is similar to a person rolling down a hill, which occurs without effort (negative work).

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