Equipotential Surfaces

1. Concept Overview / Statement of the Law

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

This means:

• No work is required to move a charge anywhere along an equipotential surface.
• Electric field is always perpendicular to an equipotential surface.
• Equipotential surfaces show how potential varies in space without having to calculate it at every point.

Equipotential surfaces help students visualize:

• Where the region of high/low potential is,
• How electric field strength changes,
• How charges interact in space,
• How potential drops in circuits and physical systems.

Just like contour lines on a map represent equal height, equipotential surfaces represent equal electric potential.

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2. Clear Explanation and Mathematical Treatment

Definition

A surface is said to be equipotential if:

[
V = \text{constant}
]

on all points of that surface.

Work Done Along Equipotential

Work done in moving a charge [q] from point A to B:

[
W = q(V_B – V_A)
]

If A and B lie on the same equipotential surface:

[
V_A = V_B \Rightarrow W = 0
]

Relation with Electric Field

Electric field is related to potential by:

[
\vec{E} = -\nabla V
]

This implies:

• Electric field lines point from higher potential to lower potential.
• Electric field is perpendicular to equipotential surfaces.

Examples of Equipotential Surfaces

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3. Dimensions and Units

Potential is measured in:

• SI Unit: Volt (V)
• Dimensions:
  [[M L^2 T^{-3} A^{-1}]]

(Electric field and surface shape have no direct dimensions here.)

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4. Key Features

1. Potential is constant at all points on an equipotential surface.
2. No work is done moving a charge along it.
3. Electric field is always perpendicular to equipotential surfaces.
4. Equipotential surfaces can never intersect.
5. Where equipotential surfaces are close together, electric field is strong.
6. Shape depends on the type of charge distribution.
7. Equipotential surfaces provide intuitive visualization of the electric field pattern.
8. Equipotentials around point charges are spherical; around dipoles are complex.

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5. Important Formulas to Remember

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6. Conceptual Questions With Solutions

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7. FAQ / Common Misconceptions

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8. Practice Questions (With Step-By-Step Solutions)

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Q1.

What work is required to move a +2 μC charge along an equipotential surface of 10 V?

Solution:

Since it is an equipotential surface:

[
V_A = V_B
]

[
W = q(V_B – V_A) = 0
]

Answer: 0 J

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Q2.

The potential difference between two equipotential surfaces is 5 V, and the distance between them is 2 mm.
Find electric field magnitude.

Solution:

[
E = -\dfrac{dV}{dr}
]

Given:

[dV] [= 5 \text{V} [,\quad] [dr = 2\times10^{-3} \text{ m}]

[E] [= \dfrac{5}{2\times10^{-3}}] [= 2500 \text{V/m}]

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Q3.

A point charge q = 6 μC creates equipotential surfaces at 20 V, 30 V, 40 V… etc.
Find the distance between 20 V and 30 V surfaces.

Solution:

Potential due to point charge:

[
V = \dfrac{1}{4\pi\varepsilon_0} \dfrac{q}{r}
]

Thus:

[
r = \dfrac{9\times 10^9 \times 6\times10^{-6}}{V}
]

For 20 V:

[r_1] [= \dfrac{54\times10^{3}}{20}] [= 2700 \text{ m}]

For 30 V:

[r_2] [= \dfrac{54\times10^{3}}{30}] [= 1800 \text{ m}]

Distance:

[
\Delta r = 900 \text{ m}
]

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Q4.

Why must electric field be perpendicular to equipotential surfaces? Show mathematically.

Solution:

Work done:

[
dW = \vec{E}\cdot d\vec{l}
]

But along equipotential:

[
dW = 0
]

Thus:

[
\vec{E}\cdot d\vec{l} = 0
]

Meaning:

[
\vec{E} \perp d\vec{l}
]

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Q5.

For a dipole, equipotential surfaces are zero on equatorial line. Explain using formula.

Solution:

Dipole potential:

[V] [= \dfrac{1}{4\pi\varepsilon_0}\dfrac{p\cos\theta}{r^2}]

On equatorial line:

[
\theta = 90^\circ \Rightarrow \cos\theta = 0
]

Thus:

[
V = 0
]