Potential due to Point Charge

Q: What is the difference between electric potential and electric potential energy?

Electric potential ($V$) is a characteristic of the electric field itself at a particular point in space, defined as the potential energy per unit positive test charge. It tells us how much potential energy a unit charge *would have* if placed there. Electric potential energy ($U$), on the other hand, is the actual energy stored by a *specific charge* $q$ when it is placed at a point where the potential is $V$. The relationship is $U = qV$. Think of potential as the 'height' of an electric hill, and potential energy as the energy a specific 'ball' (charge) has at that height.

Q: How does the electric potential vary with distance from a point charge?

For a point charge $Q$, the electric potential $V$ varies inversely with the distance $r$ from the charge, i.e., $V propto 1/r$. This means as you move further away from the charge, the magnitude of the potential decreases. In contrast, the electric field $E$ varies inversely with the square of the distance, $E propto 1/r^2$. So, potential falls off less rapidly than the electric field with increasing distance.

Updated 22 Mar 2026

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Electric potential at a point in an electric field is defined as the amount of work done by an external agent in bringing a unit positive test charge from infinity to that point without acceleration. It is a scalar quantity and is measured in Joules per Coulomb (J/C), also known as Volts (V). For a single isolated point charge $Q$ , the electric potential $V$ at a distance $r$ from the charge is gi…

Quick Summary

Electric potential due to a point charge is a fundamental concept in electrostatics, defining the 'electric state' of a point in space. It is a scalar quantity, measured in Volts (V), and represents the work done per unit positive test charge to bring it from infinity to that point without acceleration.

For an isolated point charge $Q$ , the potential $V$ at a distance $r$ is given by $V = \frac{kQ}{r}$ , where $k = \frac{1}{4piepsilon_0}$ . The sign of $Q$ is crucial: positive charges create positive potentials, and negative charges create negative potentials.

Unlike the electric field, which is a vector and varies as $1/r^2$ , potential is a scalar and varies as $1/r$ . This scalar nature simplifies calculations for multiple charges, as the total potential at a point is simply the algebraic sum of potentials due to individual charges (superposition principle).

Understanding this concept is vital for comprehending electric potential energy, work done in electric fields, and the behavior of charges in various electrostatic setups.

Full explanation

Conceptual Foundation: Work, Potential Energy, and Electric Potential

To truly grasp the concept of electric potential due to a point charge, we must first revisit the fundamental ideas of work, potential energy, and their relationship with conservative forces. The electrostatic force, like gravity, is a conservative force. This means the work done by the electrostatic force (or against it by an external agent) in moving a charge between two points depends only on the initial and final positions, not on the path taken.

When an external agent moves a test charge $q_0$ from a reference point (usually infinity, where potential is defined as zero) to a point P in an electric field, the work done against the electric force is stored as electric potential energy. If this work is $W_{infty \to P}$ , then the electric potential energy $U$ at point P is $U = W_{infty \to P}$ .

Electric potential, denoted by $V$ , is then defined as the electric potential energy per unit positive test charge. That is, $V = \frac{U}{q_0} = \frac{W_{infty \to P}}{q_0}$ . This definition makes electric potential a property of the electric field itself, independent of the test charge used to measure it. It quantifies the 'electric state' of a point in space.

Key Principles and Laws

Coulomb's Law — The foundation of electrostatics. It states that the force between two point charges

q_1

and

q_2

separated by a distance

r

is given by

F = \frac{1}{4piepsilon_0} \frac{|q_1 q_2|}{r^2}

. This force is repulsive for like charges and attractive for unlike charges. The constant

rac{1}{4piepsilon_0}

is often denoted as

k

, with a value of approximately

9 \times 10^9 , \text{N m}^2/\text{C}^2

Definition of Electric Potential — As discussed,

V = \frac{W_{infty \to P}}{q_0}

. This definition is crucial for deriving the potential due to a point charge.

Relationship between Electric Field and Potential — For a conservative field, the electric field

vec{E}

is related to the electric potential

V

by $vec{E} = -

abla V $, where$ abla $is the gradient operator. In one dimension,$ E = -rac{dV}{dr}$. This implies that the electric field points in the direction of decreasing potential.

Derivation of Potential due to a Point Charge

Let's derive the expression for the electric potential $V$ at a point P due to an isolated point charge $Q$ located at the origin. Consider a unit positive test charge $q_0$ being brought from infinity to point P, which is at a distance $r$ from $Q$ .

The electric force exerted by $Q$ on $q_0$ at any distance $x$ from $Q$ is given by Coulomb's Law:

F_e = \frac{1}{4piepsilon_0} \frac{Q q_0}{x^2}

This force is directed radially outwards if

Q

is positive. To bring

q_0

closer, an external force

F_{ext}

must be applied, equal in magnitude and opposite in direction to

F_e

, i.e.,

F_{ext} = -F_e

. We assume the charge is moved without acceleration, so

F_{net} = 0

The work done by the external agent in moving the test charge $q_0$ by an infinitesimal displacement $dvec{x}$ is $dW = vec{F}_{ext} cdot dvec{x}$ . Since we are moving the charge radially inwards, $dvec{x}$ is in the opposite direction to $vec{F}_e$ . Thus, $dW = F_{ext} dx = -F_e dx$ .

The total work done in bringing the test charge from infinity ( $infty$ ) to point P (at distance $r$ ) is:

W_{infty \to r} = int_{infty}^{r} vec{F}_{ext} cdot dvec{x} = int_{infty}^{r} left(-\frac{1}{4piepsilon_0} \frac{Q q_0}{x^2}\right) dx

Note that the displacement

dx

is negative as we move from

infty

r

Let's consider the magnitude of the force and the displacement. If we define $dx$ as a positive increment, then the force $F_{ext}$ is in the negative $x$ direction (towards $Q$ ). So, $F_{ext} = -\frac{k Q q_0}{x^2}$ .

W_{infty \to r} = int_{infty}^{r} left(-\frac{k Q q_0}{x^2}\right) dx = -k Q q_0 int_{infty}^{r} x^{-2} dx

W_{infty \to r} = -k Q q_0 left[ \frac{x^{-1}}{-1} \right]_{infty}^{r} = -k Q q_0 left[ -\frac{1}{x} \right]_{infty}^{r}

W_{infty \to r} = k Q q_0 left[ \frac{1}{r} - \frac{1}{infty} \right] = k Q q_0 left( \frac{1}{r} - 0 \right)

W_{infty \to r} = \frac{k Q q_0}{r}

By definition, the electric potential

V

at point P is the work done per unit test charge:

V = \frac{W_{infty \to r}}{q_0} = \frac{k Q q_0 / r}{q_0} = \frac{k Q}{r}

Substituting

k = \frac{1}{4piepsilon_0}

, we get:

V = \frac{1}{4piepsilon_0} \frac{Q}{r}

This is the fundamental formula for the electric potential due to a point charge

Q

at a distance

r

from it.

The sign of $Q$ must be included in the calculation. If $Q$ is positive, $V$ is positive. If $Q$ is negative, $V$ is negative.

Superposition Principle for Potential

Since electric potential is a scalar quantity, the total electric potential at a point due to a system of multiple point charges is simply the algebraic sum of the potentials due to individual charges.

This is known as the superposition principle for potential. If there are $n$ point charges $Q_1, Q_2, dots, Q_n$ at distances $r_1, r_2, dots, r_n$ respectively from a point P, the total potential at P is:

V_{total} = V_1 + V_2 + dots + V_n = sum_{i=1}^{n} \frac{k Q_i}{r_i}

This scalar addition makes calculations involving multiple charges much simpler compared to vector addition required for electric fields.

Real-World Applications

Particle Accelerators — The concept of electric potential is fundamental to how particle accelerators work. Charged particles are accelerated by moving them through regions of varying electric potential, gaining kinetic energy as they move from higher to lower potential (for positive charges) or vice-versa (for negative charges).

Capacitors — Capacitors store electric potential energy by creating a potential difference between two conducting plates. The potential difference is directly related to the charge stored on the plates.

Electrostatic Precipitators — Used to remove particulate matter from industrial exhaust gases. High potential differences are used to charge particles, which are then attracted to oppositely charged collection plates.

Biological Systems — Nerve impulses involve changes in electric potential across cell membranes, crucial for communication within the body.

Common Misconceptions

Potential vs. Potential Energy — Students often confuse electric potential (

V

) with electric potential energy (

U

). Potential is potential energy per unit charge (

V = U/q_0

), a property of the field itself. Potential energy is the energy stored by a specific charge in that field (

U = q_0 V

Sign Convention — Forgetting to include the sign of the source charge

Q

in the potential calculation. A positive charge creates positive potential, and a negative charge creates negative potential. This is critical for correct algebraic summation.

Vector vs. Scalar — Mistaking potential for a vector quantity. Electric potential is a scalar; it has magnitude and sign but no direction. This is a key difference from the electric field, which is a vector.

Dependence on Path — Believing that the work done (and thus potential) depends on the path taken. Since the electrostatic force is conservative, the work done in moving a charge between two points is path-independent.

Reference Point — Not understanding that potential is always defined relative to a reference point. While infinity is the standard reference for isolated charges, other reference points can be chosen, leading to different absolute potential values but the same potential difference.

NEET-Specific Angle

For NEET, understanding potential due to a point charge is foundational. Questions often involve:

Direct calculation — Applying

V = kQ/r

for single or multiple charges.

Superposition principle — Calculating total potential at a point due to a system of charges (e.g., at the center of a square, triangle, or along an axis).

Work done — Relating potential difference to work done:

W_{AB} = q(V_B - V_A)

. This is a very common question type.

Equipotential surfaces — Understanding that for a point charge, equipotential surfaces are concentric spheres. No work is done in moving a charge along an equipotential surface.

Relationship with Electric Field — Conceptual questions about how

V

changes with

r

compared to

E

(i.e.,

V propto 1/r

vs.

E propto 1/r^2

). Also, the direction of

E

being from higher to lower potential.

Graphical representation — Interpreting

V

vs.

r

graphs for positive and negative point charges.

Potential energy of a system — Calculating the potential energy of a system of point charges by bringing them one by one from infinity. This involves summing

q_i V_i

where

V_i

is the potential due to all other charges at the position of

q_i

Key Concepts

Derivation from Work Done

The potential at a point is derived by calculating the work done by an external agent to move a unit positive…

Superposition Principle for Potential

Since potential is a scalar, the total potential at any point due to a system of point charges is the…

Potential Difference and Work Done

The potential difference $Delta V = V_B - V_A$ between two points A and B is the work done per unit positive…

Often confused with

Side-by-side differences the NEET paper likes to test.

Potential due to Point Charge vs Electric Field due to Point Charge

Aspect	Potential due to Point Charge	Electric Field due to Point Charge
Nature	Scalar quantity (magnitude only)	Vector quantity (magnitude and direction)
Formula	$V = \frac{kQ}{r}$	$E = \frac{k\|Q\|}{r^2}$ (magnitude)
Dependence on distance (r)	Varies as $1/r$	Varies as $1/r^2$
Sign	Can be positive (for +Q) or negative (for -Q)	Magnitude is always positive; direction depends on sign of Q (radially outward for +Q, inward for -Q)
Superposition	Algebraic sum (scalar addition)	Vector sum (requires components)
Units	Volt (V) or J/C	N/C or V/m

Electric potential and electric field, though intimately related, are distinct physical quantities. Potential is a scalar measure of the 'energy landscape' per unit charge, varying inversely with distance ( $1/r$ ).

It can be positive or negative, reflecting the nature of the source charge. The electric field, conversely, is a vector quantity representing the force per unit charge, varying inversely with the square of the distance ( $1/r^2$ ).

Its direction indicates the force on a positive test charge. Due to their scalar vs. vector nature, calculations involving multiple charges are significantly simpler for potential (algebraic sum) than for the electric field (vector sum).

Why it is tested: For NEET, distinguishing between electric potential and electric field is critical. Questions frequently test understanding of their scalar/vector nature, their dependence on distance, and how to apply the superposition principle for each. Misconceptions often arise from confusing their formulas or their methods of summation. A clear understanding of their differences helps in solving both conceptual and numerical problems accurately, especially when comparing their behavior in various configurations of charges.

Questions students ask

6 answered on this topic.

What is the difference between electric potential and electric potential energy?

Electric potential ( $V$ ) is a characteristic of the electric field itself at a particular point in space, defined as the potential energy per unit positive test charge. It tells us how much potential energy a unit charge would have if placed there.

Electric potential energy ( $U$ ), on the other hand, is the actual energy stored by a specific charge $q$ when it is placed at a point where the potential is $V$ . The relationship is $U = qV$ . Think of potential as the 'height' of an electric hill, and potential energy as the energy a specific 'ball' (charge) has at that height.

Why is electric potential a scalar quantity while electric field is a vector quantity?

Electric potential is defined based on the work done, which is a scalar quantity. Work involves the dot product of force and displacement, resulting in a scalar value. Since potential is work done per unit charge, it also remains a scalar.

Electric field, however, is defined as the force per unit charge, and force is a vector quantity (having both magnitude and direction). Therefore, the electric field inherits the vector nature of force, indicating the direction in which a positive test charge would experience a force.

Why do we take infinity as the reference point for zero potential?

Choosing infinity as the reference point for zero potential is a convention that simplifies calculations, especially for isolated charge distributions that extend over finite regions. At infinite distance, the electric force due to a finite charge distribution becomes negligible, making the work done to bring a charge from infinity a convenient and well-defined measure of potential. While other reference points can be chosen, infinity provides a universal and consistent baseline.

How does the electric potential vary with distance from a point charge?

For a point charge $Q$ , the electric potential $V$ varies inversely with the distance $r$ from the charge, i.e., $V propto 1/r$ . This means as you move further away from the charge, the magnitude of the potential decreases. In contrast, the electric field $E$ varies inversely with the square of the distance, $E propto 1/r^2$ . So, potential falls off less rapidly than the electric field with increasing distance.

What is an equipotential surface for a point charge?

An equipotential surface is a surface over which the electric potential is constant. For a single isolated point charge, the equipotential surfaces are concentric spheres centered on the charge. This is because the potential $V = kQ/r$ depends only on the distance $r$ . Any point at the same distance $r$ from the point charge will have the same potential. Moving a test charge along an equipotential surface requires no work, as there is no change in potential energy.

Can electric potential be negative? What does it signify?

Yes, electric potential can be negative. If the source charge $Q$ is negative, then the potential $V = kQ/r$ will be negative. A negative potential signifies that positive work would be done by the electric field (or negative work by an external agent) to bring a positive test charge from infinity to that point. In essence, the field 'pulls' the positive test charge in, meaning it naturally moves towards the negative source charge, losing potential energy in the process.

Revise in 30 seconds

Definition: — Work done by external agent to bring unit positive charge from infinity to a point.

Formula: —

V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} = \frac{kQ}{r}

Units: — Volt (V) or J/C

Nature: — Scalar quantity (includes sign of Q)

Sign: — Positive for +Q, Negative for -Q

Dependence: —

V \propto 1/r

Superposition: —

V_{total} = \sum V_i = \sum \frac{kQ_i}{r_i}

(algebraic sum)

Work Done: —

W_{ext} = q(V_{final} - V_{initial})

Relationship with E: —

E = -dV/dr

(in 1D),

r = V/E

(for point charge)

Equipotential Surfaces: — Concentric spheres for point charge;

W=0

along them.

To remember the potential formula and its properties:

Very Positive Charges Radiate Potential, Negative Charges Attract Negative Potential.

Very: Voltage (Potential)

Positive Charges:

Q

is positive

Radiate Potential:

V

is positive

Negative Charges:

Q

is negative

Attract Negative Potential:

V

is negative

And for the formula: Very Kool Quick Review: $V = kQ/r$