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Electric Potential

CSCA Electric Potential study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Topic: Electromagnetism > Electrostatics > Electric Potential

1. Core Concept: Electric Potential ($V$)

**Electric Potential** is a scalar quantity describing the **energy attribute** of an electric field.

To understand it intuitively, use the **Gravity Analogy**:

* **Electric Potential** $\approx$ **Height** ($h$)

* **Electric Field Lines** $\approx$ **Slope Direction**

* **Positive Charge** creates a "**Hill**" (High Potential).

* **Negative Charge** creates a "**Valley**" (Low Potential).

**Definition**:

The work done by the electric force in moving a unit positive charge from a point to the **Zero Potential Reference** (usually infinity).

* **Formula**: $V = \frac{E_p}{q} = \frac{W}{q}$

* **Unit**: Volt ($V$), $1V = 1 J/C$.

* **Scalar**: It has magnitude and sign (+/-), but NO direction.

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2. Core Formulas

(1) Potential of a Point Charge

For a stationary point charge in a vacuum ($V_{\infty}=0$).

$$V = k \frac{Q}{r}$$

**CSCA Exam Warning**:

Unlike Coulomb's Force calculation, you **MUST include the sign (+/-)** of the charge $Q$ in this formula!

* Potential around a positive charge is **positive**.

* Potential around a negative charge is **negative**.

(2) Superposition Principle

The total potential is the **Algebraic Sum** of potentials produced by individual charges.

$$V_{total} = V_1 + V_2 + ...$$

(Simple addition of numbers, no vector decomposition needed).

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3. Potential Difference & Work

**Potential Difference (Voltage)**:

$$U_{AB} = V_A - V_B$$

**Work Done by Electric Force**:

$$W_{AB} = q U_{AB} = q(V_A - V_B)$$

* **Key Rules**:

* **Along the electric field line, Potential decreases.**

* Positive charges naturally move from High $V$ $\to$ Low $V$.

* Negative charges naturally move from Low $V$ $\to$ High $V$.

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4. Equipotential Surfaces

Surfaces where the potential is equal everywhere. Similar to "Contour Lines" on a map.

**Four Properties**:

1. **Perpendicular**: Always perpendicular to Electric Field lines.

2. **Zero Work**: Moving a charge along an equipotential surface requires zero work.

3. **Direction**: Field lines point from Higher Potential surfaces to Lower ones.

4. **Density**: Dense equipotential lines indicate a stronger Electric Field ($E$).

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5. Typical Examples

**Example 1: Calculation (Watch the Sign)**

**Problem**: A point charge $Q = -3.0 \times 10^{-9} C$. Find $V$ at $r = 0.1 m$.

**Solution**:

Substitute with sign:

$$V = k \frac{Q}{r} = 9.0 \times 10^9 \times \frac{-3.0 \times 10^{-9}}{0.1} = -270 \, V$$

**Example 2: Superposition (Dipole)**

**Problem**: At the midpoint between $+Q$ and $-Q$, what are the Potential and Field Strength?

**Solution**:

* **Potential**: $V_{mid} = V_+ + V_- = k\frac{Q}{r} + k\frac{-Q}{r} = 0$. (Scalar sum is zero).

* **Field**: Directions are the same, so they add up. $E_{mid} \neq 0$.

**Example 3: Work Calculation**

**Problem**: Move an electron ($q = -1.6\times 10^{-19}C$) from $A$ ($10V$) to $B$ ($6V$). Calculate work.

**Solution**:

$$W_{AB} = q(V_A - V_B) = (-1.6\times 10^{-19}) \times (4) = -6.4 \times 10^{-19} \, J$$