Potential due to a System of Charges & Equipotential Surfaces

(Class 12 Physics – NCERT Notes)

---

1. Recap: Electrostatic Potential

Electrostatic potential (V) at a point is defined as:

The work done by an external agent in bringing a unit positive test charge from infinity to that point, without acceleration.

$$V=\frac{W}{q}$$

• SI unit: volt (V)

• 1 volt = 1 joule / coulomb

---

2. Potential Due to a System of Charges

What is a system of charges?

A system of charges consists of two or more point charges placed at different positions in space.

Principle Used: Superposition Principle

The electrostatic potential due to a system of charges is the algebraic sum of potentials due to individual charges.

⚠️ Important:

• Potential is a scalar quantity

• So, we add directly, not vectorially

---

Formula: Potential at a Point Due to n Charges

Suppose there are charges
$q_1,q_2,q_3,\dots ,q_n$
at distances
$r_1,r_2,r_3,\dots ,r_n$
from a point P.

$$V=\frac{1}{4\pi {\epsilon }_0}(\frac{q_1}{r_1}+\frac{q_2}{r_2}+\frac{q_3}{r_3}+\cdots +\frac{q_n}{r_n})$$

---

Key Points to Remember

• Potential may be positive, negative, or zero

• Zero potential does not mean zero electric field

• Reference potential is usually taken as zero at infinity

---

Special Cases

1. Like charges → potentials add up

2. Unlike charges → potentials may cancel

3. At midpoint of equal and opposite charges → potential can be zero

---

3. Potential Due to a Continuous Charge Distribution

If charge is continuously distributed (line, surface, or volume):

$$V=\frac{1}{4\pi {\epsilon }_0}\int \frac{dq}{r}$$

Where:

• $dq$ = small charge element

• $r$ = distance of $dq$ from the point

---

4. Equipotential Surfaces

Definition

An equipotential surface is a surface on which the electric potential is same at every point.

Important Properties of Equipotential Surfaces

1. No work is done in moving a charge along an equipotential surface

   $$W=q(V_1-V_2)=0$$

2. Electric field is always perpendicular to equipotential surfaces

   • If it were not, work would be done → contradiction

3. Equipotential surfaces never intersect

   • One point cannot have two different potentials

4. Closer surfaces → stronger electric field

---

5. Equipotential Surfaces for Different Charge Configurations

(a) Point Charge

• Shape: Concentric spheres

• Center: at the charge

• Potential depends only on distance $r$

---

(b) Uniform Electric Field

• Shape: Parallel planes

• Perpendicular to direction of field

---

(c) Electric Dipole

• Complex curved surfaces

• Symmetric about dipole axis

• Equatorial plane is an equipotential surface with zero potential

---

6. Relation Between Electric Field and Potential

Electric field is the negative gradient of potential:

$$E=-\frac{dV}{dl}$$

Meaning

• Electric field points in the direction of maximum decrease of potential

• Greater the rate of change of potential → stronger the field

---

7. Comparison: Electric Field vs Potential

---

8. Common Exam Mistakes (Very Important ⚠️)

• ❌ Adding potentials vectorially

• ❌ Saying work is done on equipotential surface

• ❌ Confusing zero potential with zero field

• ❌ Forgetting reference potential (usually infinity)

---

9. NCERT-Style One-Line Answers

• Potential due to a system of charges is the algebraic sum of individual potentials.

• Equipotential surfaces are always perpendicular to electric field lines.

• Work done on an equipotential surface is zero.

• Electric field is directed from higher to lower potential.

---

10. Numerical Tip for Boards

Always:

1. Write formula first

2. Mention reference potential

3. Keep sign of charge carefully

4. Use SI units