Electrostatic Potential & Potential Difference

Introduction

In the previous chapter, we studied electric charges and the forces they exert (Coulomb's Law) and the fields they create ($\vec{E}$). These are vector quantities, which can become mathematically complex when dealing with many charges.

Electrostatic Potential ($V$) provides a scalar way to describe the same physics. It is related to the work done and energy in an electric field. Just as a ball flows from high altitude to low altitude due to gravity, positive charges 'flow' from high electric potential to low electric potential.

Conservative Nature of Electrostatic Force

Before defining potential, we must understand that the electrostatic force is a conservative force. This means:

1. The work done by an electrostatic field in moving a charge from one point to another depends only on the initial and final positions, not on the path taken.
2. The work done in moving a charge over a closed loop is zero ($\oint \vec{E} \cdot d\vec{l} = 0$).

Electrostatic Potential at a Point

Defining Potential at a Point

To understand Electrostatic Potential ($V$) at a specific point, imagine an electric field created by a source charge $+Q$. If you place a tiny test charge $q_0$ at any point in this field, it experiences a force. To hold it there or move it, work must be done.

Formal Definition: The electrostatic potential at a point is the work done by an external force in bringing a unit positive charge from infinity to that point, without any acceleration (i.e., moving it infinitely slowly).

Why Without Acceleration?

In Physics, if an object accelerates, its kinetic energy ($K$) changes. We want the work done to be stored entirely as potential energy ($U$). By moving the charge at a constant, vanishingly small speed, we ensure that: $$W_{ext} = \Delta U$$

The Reference Point: Infinity

We define the potential at infinity as zero ($V_{\infty} = 0$). This is because at an infinite distance, the electric field of any local source charge becomes zero, meaning no force is exerted and no work is needed to move a charge there.

Mathematical Representation

If $W_{\infty \to P}$ is the work done in bringing a test charge $q_0$ from infinity to point $P$, then the potential $V$ at point $P$ is:

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

Important Properties:

• Scalar Nature: Unlike the Electric Field ($\vec{E}$), Potential ($V$) is a scalar. It has no direction. If you have multiple source charges, you simply add their potentials algebraically ($V_{total} = V_1 + V_2 + \dots$).
• Sign of Potential:
  • Potential is positive if the source charge is positive (work is done against repulsion).
  • Potential is negative if the source charge is negative (the field 'pulls' the charge in, and the external agent does 'negative' work to keep it from accelerating).

Physical Significance

Think of Electrostatic Potential as Electrical Level.

• Positive charges naturally move from higher potential to lower potential (like water flowing downhill).
• Negative charges (like electrons) move from lower potential to higher potential (like a bubble rising in water).

SI Unit and Dimensions

• Unit: Volt (V). $1 \text{ V} = 1 \text{ Joule/Coulomb}$. One Volt is defined as 1 Joule of work done per 1 Coulomb of charge.
• Dimensions: Potential $= \frac{\text{Work}}{\text{Charge}} = \frac{[ML^2T^{-2}]}{[AT]} = [ML^2T^{-3}A^{-1}]$.

Electrostatic Potential Difference ($\Delta V$)

Definition: The electrostatic potential difference between two points $A$ and $B$ in an electric field is the amount of work done by an external agent in moving a unit positive test charge from $A$ to $B$ without acceleration (slowly).

Derivation: Consider a source charge $+Q$ at the origin. To move a test charge $q_0$ from point $A$ to point $B$ against the repulsive force $\vec{F}_{elec}$:

1. The external force required is $\vec{F}_{ext} = -\vec{F}_{elec} = -q_0 \vec{E}$.
2. The small work done for a displacement $d\vec{r}$ is $dW = \vec{F}_{ext} \cdot d\vec{r}$.
3. The total work done from $A$ to $B$ is: $$W_{AB} = \int_{A}^{B} \vec{F}_{ext} \cdot d\vec{r} = -q_0 \int_{A}^{B} \vec{E} \cdot d\vec{r}$$
4. By definition, $\Delta V = V_B - V_A = \frac{W_{AB}}{q_0}$: $$V_B - V_A = -\int_{A}^{B} \vec{E} \cdot d\vec{r}$$

🧠 Memory Capsule

• Work-Energy Link: Work done by the field $= -\Delta U$. Work done by external agent $= +\Delta U$.
• Reference: Potential is relative. Always remember $V_{\infty} = 0$.
• Scalar Advantage: No need to worry about directions or components when adding potentials from different charges—just simple algebra!
• High to Low: Positive charges move from high $V$ to low $V$. Negative charges (like electrons) move from low $V$ to high $V$.
• The Formula: $V = W/q$. Always check if $W$ is work done by the external agent (usually $+$) or the field (usually $-$).

Example 1: Calculating Work from Potential

How much work is required to bring a $2 \mu C$ charge from infinity to a point where the potential is $10^5$ V?

Solution:

1. Given: $q = 2 \times 10^{-6}$ C, $V = 10^5$ V.
2. Formula: $$W = qV$$
3. Calculation: $$W = (2 \times 10^{-6}) \times 10^5 = 0.2\text{ J}$$
4. Final Answer: $$W = 0.2\text{ J}$$

Example 2: Negative Work and Negative Potential

A $5$ C charge is brought from infinity to a point where the potential is $-20$ V. Find the work done by the external agent.

Solution:

1. Given: $q = 5$ C, $V = -20$ V.
2. Formula: $$W = qV$$
3. Calculation: $$W = 5 \times (-20) = -100\text{ J}$$
4. Meaning: The negative sign means the external agent does negative work because the field itself helps bring the charge in.
5. Final Answer: $$W = -100\text{ J}$$

Example 3: Potential Energy from Potential

A point $P$ has a potential of $50$ V. If a charge of $-4 \mu C$ is placed at $P$, what is its electrostatic potential energy?

Solution:

1. Given: $V = 50$ V, $q = -4 \times 10^{-6}$ C.
2. Relation: $$U = qV$$
3. Calculation: $$U = (-4 \times 10^{-6}) \times 50 = -2 \times 10^{-4}\text{ J}$$
4. Final Answer: $$U = -2 \times 10^{-4}\text{ J}$$

Example 4: Finding Potential from Work

If $20$ J of work is done in bringing a $2$ C charge from infinity to point A, what is the potential at A?

Solution:

1. Given: $$W_{\infty \to A} = 20\text{ J}, \quad q = 2\text{ C}$$
2. Formula: $$V_A = \frac{W_{\infty \to A}}{q}$$
3. Calculation: $$V_A = \frac{20}{2} = 10\text{ V}$$
4. Final Answer: $$V_A = 10\text{ V}$$

Example 5: Work Done by the Field

To move a charge $Q$ from point A to B, $15$ J of work is done by the electric field. Find the potential difference $V_B - V_A$.

Solution:

1. Work done by field and potential difference relation: $$W_{field} = -Q(V_B - V_A)$$
2. Substitute: $$15 = -Q(V_B - V_A)$$
3. Rearrange: $$V_B - V_A = -\frac{15}{Q}$$
4. Final Answer: $$V_B - V_A = -\frac{15}{Q}\text{ V}$$

Example 6: Comparing Potentials at Different Distances

Two points $X$ and $Y$ are at distances $r$ and $2r$ from a point charge $+Q$. At which point is the potential higher?

Solution:

1. Using point-charge potential: $$V = \frac{kQ}{r}$$
2. For a positive charge, potential decreases as distance increases.
3. Since $r < 2r$, the point at distance $r$ has higher potential.
4. Final Answer: Point $$X$$ has the higher potential.

Example 7: Path Independence

A charge is moved from $(0,0)$ to $(3,4)$ in an electrostatic field. Path 1 is a straight line. Path 2 is via $(3,0)$. Compare the work done.

Solution:

1. Electrostatic force is conservative.
2. Therefore, work done depends only on the initial and final positions.
3. Since both paths have the same start and end points, the work done is the same.
4. Final Answer: $$W_{\text{Path 1}} = W_{\text{Path 2}}$$

Example 8: Work Done on a Closed Path

What is the work done in moving a test charge once around a closed path in an electrostatic field?

Solution:

1. For electrostatic forces, work done over a closed loop is zero.

2. Equivalently, the initial and final points are the same, so there is no change in potential.

3. Therefore, $W = 0$

4. Final Answer: $W = 0$

Example 9: Direction of Electron Motion

Between two points A ($V = 10$ V) and B ($V = 5$ V), which way will an electron move spontaneously?

Solution:

1. Negative charges move from lower potential to higher potential.
2. Here, B is at lower potential and A is at higher potential.
3. Therefore, the electron moves from B to A.
4. Final Answer: The electron moves from B to A.