Potential due to Point Charge — Explained

Detailed 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

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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 $9 \times 10^9 , \text{N m}^2/\text{C}^2$.
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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.
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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 $infty$ to $r$.

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

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

Common Misconceptions

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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 ($U = q_0 V$).
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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.
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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.
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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.
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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 $q_i$.